Id like to see some numbers supporting your claim that blade pact is greater damage than eldtritch blast, please. Can you give me like statuses at level 1, 3, 5 and maybe even 12?
At first level neither exists, at 3rd level the difference is 2d6 base vs 1d10 base for a slight advantage. Unless a +1 weapon is available or human variant with a feat has been taken (like pole arm master). At 4th level a feat is available regardless.
At 11th level that 3rd beam appears before the lifedrinker invocation at 12th level would be available for the blade pact and that can be impacting. Here are some numbers I posted at Giantip and you can ask if you have something specific you want to know.
At first level, TWF does more damage than eldritch blast. At second and third levels agonizing blast becomes available and the damage is still better using TWF because hex applies to both attacks. At fourth level the option is available for a feat or an ability increase. Great weapon master can be taken, or pole arm master can be taken. Caster would take CHA. This is what it looks like vs AC 16 with hex up and 18 CHA vs 16 STR, for example:
Eldritch Blast (with agonizing and hex):
5% crit chance -- 1d10+1d10+4+1d6+1d6 for 1.1
50% standard -- 1d10+4+1d6 for 6.5
45% miss chance -- nada
At one attack 7.6 avg damage.
Great Sword (with heavy weapon master used)
5% crit chance -- 2d6+2d6+3+10+d6+d6 plus bonus attack for the crit for 2.02
20% standard -- 2d6+10+3+1d6 for 4.7
75% miss chance -- nada
At one attack for 6.72 avg damage and less than eldritch blast at this point.
Great Sword (with heavy weapon master not used because the feat can simply be taken early and used on lower AC opponents)
5% crit chance -- 2d6+2d6+3+1d6+1d6 plus bonus attack for the crit for 1.56375
45% standard -- 2d6+3+1d6 for 6.075
50% miss chance -- nada
At one attack for 7.63875 and almost identical to eldritch blast, very slightly better, with possible leverage on low AC targets.
Halberd (with pole arm master)
Main Attack
5% crit chance -- 1d10+1d10+3+1d6+1d6 for 1.05
45% standard -- 1d10+3+1d6 for 5.4
50% miss chance -- nada
Bonus Attack
5% crit chance -- 1d4+1d4+3+1d6+1d6 for 0.75
45% standard -- 1d4+3+1d6 for 4.05
50% miss chance -- nada
For both attacks 11.25 avg damage and better than eldritch blast or great weapon master.
There doesn't appear to be an issue where weapons are less damage than eldritch blast after taking accuracy into consideration when the feats become available at 4th level. At 5th level the second blast appears and the blade warlock gains access to the thirsting blade invocation for an extra attack.
Eldritch Blast (with agonizing and hex):
5% crit chance -- 1d10+1d10+4+1d6+1d6 for 1.1
55% standard -- 1d10+4+1d6 for 7.15
40% miss chance -- nada
At two attacks 16.5 avg damage for a nice jump.
Great Sword (with heavy weapon master used)
5% crit chance -- 2d6+2d6+3+10+d6+d6 plus bonus attack for the crit for 2.07875 or 1.7 (crit can only give 1 bonus attack)
25% standard -- 2d6+10+3+1d6 for 5.875
70% miss chance -- nada
At two attacks for 15.52875 avg damage (and a miniscule amount more but I didn't both with chance for crit twice) and still less than eldritch blast at this point.
Great Sword (with heavy weapon master not used because the feat can simply be taken early and used on lower AC opponents)
5% crit chance -- 2d6+2d6+3+1d6+1d6 plus bonus attack for the crit for 1.5975 or 1.2 (crit can only give 1 bonus attack)
50% standard -- 2d6+3+1d6 for 6.75
45% miss chance -- nada
At two attacks for 16.2975 avg damage (and a miniscule amount more but I didn't both with chance for crit twice) and almost eldritch blast with possible leverage on low AC targets.
Halberd (with pole arm master)
Main Attack
5% crit chance -- 1d10+1d10+3+1d6+1d6 for 1.05
50% standard -- 1d10+3+1d6 for 6.0
45% miss chance -- nada
Bonus Attack
5% crit chance -- 1d4+1d4+3+1d6+1d6 for 0.75
50% standard -- 1d4+3+1d6 for 4.5
45% miss chance -- nada
For all attacks 19.35 avg damage and better than eldritch blast or great weapon master.
There doesn't appear to be an issue where weapons are less damage than eldritch blast after taking accuracy into consideration when the extra attacks become available at fifth level. The crit chance bonus attack loses a bit because it can only be done once regardless and great weapon master is pretty similar in damage (better if a magic weapon becomes available). Pole arm master is still the better option. At 8th level both feats can be combined while number of attacks remains the same, and the eldritch blast caster can jump up to 20 CHA.
Eldritch Blast (with agonizing and hex):
5% crit chance -- 1d10+1d10+5+1d6+1d6 for 1.15
60% standard -- 1d10+5+1d6 for 8.4
35% miss chance -- nada
At two attacks 19.1 avg damage for a small jump and still behind pole arm master with a halberd at 5th level.
Halberd (with pole arm master and great weapon master using -5/+10)
Main Attack
5% crit chance -- 1d10+1d10+3+10+1d6+1d6 for 1.55
25% standard -- 1d10+3+10+1d6 for 5.5
70% miss chance -- nada
Bonus Attack non crit main attacks (90.25%)
5% crit chance -- 1d4+1d4+3+10+1d6+1d6 for 1.25
25% standard -- 1d4+3+10+1d6 for 4.75
70% miss chance -- nada
Bonus Attack crit main attacks (9.75% chance)
5% crit chance -- 1d10+1d10+3+10+1d6+1d6 for 1.55
25% standard -- 1d10+3+10+1d6 for 5.5
70% miss chance -- nada
For all attacks 20.202375 avg damage and better than eldritch blast, and a slight bump up.
Halberd (with pole arm master and great weapon master not using -5/+10)
Main Attack
5% crit chance -- 1d10+1d10+3+1d6+1d6 for 1.05
50% standard -- 1d10+3+1d6 for 6
45% miss chance -- nada
Bonus Attack non crit main attacks (90.25%)
5% crit chance -- 1d4+1d4+3+1d6+1d6 for 0.75
50% standard -- 1d4+3+1d6 for 4.5
45% miss chance -- nada
Bonus Attack crit main attacks (9.75% chance)
5% crit chance -- 1d10+1d10+3+1d6+1d6 for 1.05
50% standard -- 1d10+3+1d6 for 6
45% miss chance -- nada
For all attacks 19.5255 avg damage and better than eldritch blast, and slightly lower than applying both feats.
At eighth level, the feats are still the superior choice due to the ability to run 3 attacks first. Levels 12, 16, and 19 can be used to apply ability modifiers for the weapons. At level eleven, eldritch blast gets a third attack while life drinker isn't available until twelth level. At level eleven, it looks like this.
Eldritch Blast (with agonizing and hex):
5% crit chance -- 1d10+1d10+5+1d6+1d6 for 1.15
65% standard -- 1d10+5+1d6 for 9.1
30% miss chance -- nada
At three attacks 30.75 avg damage for a good jump.
Halberd (with pole arm master and great weapon master using -5/+10)
Main Attack
5% crit chance -- 1d10+1d10+3+10+1d6+1d6 for 1.55
30% standard -- 1d10+3+10+1d6 for 6.6
65% miss chance -- nada
Bonus Attack non crit main attacks (90.25%)
5% crit chance -- 1d4+1d4+3+10+1d6+1d6 for 1.25
30% standard -- 1d4+3+10+1d6 for 5.7
65% miss chance -- nada
Bonus Attack crit main attacks (9.75% chance)
5% crit chance -- 1d10+1d10+3+10+1d6+1d6 for 1.55
30% standard -- 1d10+3+10+1d6 for 6.6
65% miss chance -- nada
For all attacks 23.367 avg damage and a bump up but finally behind eldritch blast; at least ignoring the likelihood of a magical weapon by this point.
So far, most levels favor weapon combat given similar bonuses. Eleventh level is a nice boost with third attack for eldritch blast but level twelve is another boost for the blade warlock. At level twelve it looks more like this:
Eldritch Blast (with agonizing and hex):
5% crit chance -- 1d10+1d10+5+1d6+1d6 for 1.15
65% standard -- 1d10+5+1d6 for 9.1
30% miss chance -- nada
At three attacks 30.75 avg damage and no change from eleventh level.
Halberd (with pole arm master and great weapon master using -5/+10)
Main Attack
5% crit chance -- 1d10+1d10+4+2+10+1d6+1d6 for 1.7
35% standard -- 1d10+4+2+10+1d6 for 8.75
60% miss chance -- nada
Bonus Attack non crit main attacks (90.25%)
5% crit chance -- 1d4+1d4+4+2+10+1d6+1d6 for 1.4
35% standard -- 1d4+4+2+10+1d6 for 7.7
60% miss chance -- nada
Bonus Attack crit main attacks (9.75% chance)
5% crit chance -- 1d10+1d10+4+2+10+1d6+1d6 for 1.7
35% standard -- 1d10+4+2+10+1d6 for 8.75
60% miss chance -- nada
For all attacks 30.131625 avg damage having added +2 damage for CHA bonus and using the ability increase for STR, back ahead of eldritch blast and still assuming no magic pole arm.
Let's skip ahead to 17th level because there aren't a lot of big jumps until then. Giving STR and the next ability increase again for the blade warlock, foresight for both, and a 4th attack for eldritch blast it looks more like this:
Eldritch Blast (with agonizing and hex):
9.75% crit chance -- 1d10+1d10+5+1d6+1d6 for 2.2425
74.25% standard -- 1d10+5+1d6 for 10.395
4% miss chance -- nada
At four attacks for 50.55 avg damage and rather nice damage.
Halberd (with pole arm master and great weapon master using -5/+10)
Main Attack
9.75% crit chance -- 1d10+1d10+5+2+10+1d6+1d6 for 3.4125
70% standard -- 1d10+5+2+10+1d6 for 18.2
20.25% miss chance -- nada
Bonus Attack non crit main attacks (81.450625%)
9.75% crit chance -- 1d4+1d4+5+2+10+1d6+1d6 for 2.8275
70% standard -- 1d4+5+2+10+1d6 for 16.24
20.25% miss chance -- nada
Bonus Attack crit main attacks (18.549375% chance)
9.75% crit chance -- 1d10+1d10+5+2+10+1d6+1d6 for 3.4125
70% standard -- 1d10+5+2+10+1d6 for 18.2
20.25% miss chance -- nada
For all attacks 62.76458159375 avg damage and still ahead of eldritch blast and still assuming no magic pole arm.
This allows for moderately armored on a variant human at first level. Pole arm mastery doesn't list the bonus attack as anything other than a 1d4 bonus attack so no restriction for ability modifier damage applies and the 3 attacks early gives a good advantage in damage.