Calculate the average turn order a ranged attack would have gone at under the old system, which included the full dex bonus. Now compare it to this new system, which does not include their dex bonus. I think you will find they're going roughly the same point in combat that they always were, relative to the str-melee combatant. I don't think they're really made more powerful - they're just about the same amount of powerful from this system I suspect.
So just as a guesstimate, a two-handed fighter would typically wear heavy armor and dump Dex, as it's only purpose would be for Initiative, and dex saves. Initiative isn't enough, on it's own, to incentivize that player to increase their dex over Con or the all important Wisdom save, and Dex saves tend to be the least impactful for that fighter (because they impact hit points, which they tend to have more of anyway). So, let's assume the Fighter has an 8 or 10 in Dex (-1 or 0), which I think is fair.
The Ranged Attacker, however, will max dex. Their AC depends on it, their attack bonus does, and their damage does. So they will likely have a 16 to 18 dex (+3 or +4), and eventually a 20 (+5).
Old system: Ranged attacker rolls a d20+3 or 4, so an average of 13.5 or 14.5. Melee attacker rolls a d20-1 or +0, so an average roll of 9.5 or 10.5. It's roughly a 4 point bonus the ranged attacker gets, and eventually a 5 point advantage once the ranged attacker gets a 20 in dex.
New system: Ranged attack rolls a d4, so average roll of 2.5. Melee attacker rolls a d8, so average of 4.5. So the Ranged attacker has a +2 bonus on the Melee attacker.
I am not sure of the percentages on that second one, but to me it seems roughly pretty darn similar. Seems like the Ranged attacker has roughly a 20-25% advantage on initiatives in both systems.
Your math is close, but your example is limited.
A 5-point advantage in initiative translates into about a 75% chance of going first, yes. Calculation: If the faster fighter rolls a 1 on initiative, the slower needs to roll a 7+ to go faster (breaking ties in favor of higher bonus). That's 14 out of 20. Extending that across the entire range, the slower fighter has a (1+2+...+14)/400 chance of going faster, or 105/400 = ~25%.
And just comparing the ranged d4 to the melee d8, the melee fighter (let's call her Brienne) has a 3/16 chance of going faster (ties break in favor of higher Dex, meaning the ranged guy - let's call him Lolegolas). That's about 17%. So the GH initiative gives Lolegolas a higher chance for winning initiative over Brienne than the core rules, but not overly so.
But that's not the whole story. Lolegolas will almost certainly be able to make his attacks, because (a) he likely goes first, and (b) even if people move around, he's likely to be able to attack someone. So he makes like a turret: standing still and shooting. But Brienne is not so lucky, as shown by Rath's frustration in the examples given. If Lolegolas drops her opponent, or if her opponent runs away, she can
waste her turn, and nobody likes that. So she always wants to combine her melee with a move, and now she has to beat Lolegolas's d4 with her own d6+d8. The chance of her doing that is 3/(4*6*8), or 1/64. That's close enough to zero that you might as well just say "ranged go first, then people move, then they fight, and then they cast spells."
You're also assuming that every melee person is a heavily armored character who eschews Dexterity. That's not really true. There are plenty of melee warriors who do not fit that description: rogues, monks, Dex-based fighters, many valor bards, barbarians, dual-wielding rangers. Under the regular rules, a monk (let's call her Ember, for old times' sake) is likely to have as high a Dex as the archer, so their initiatives should be roughly 50/50. But under GH initiative, the monk is heavily penalized - the monk will likely both move, attack, and then attack again as a bonus action - that's an initiative of d6+d8+d8. In order to win initiative, Ember will either have to straight-up roll lower than Lolegolas (1/1536), or roll a tie (4/1536) which is resolved in her favor 50% of the time (so 2/1536, for a total of 3/1536 = 1/512). Going from 50% chance of winning initiative to ~0.2% sounds like a harsh penalty to me.
