No it is not, because the first roll has to succeed in order to use it. This is a statistical fallacy. Once the first roll is already rolled (and suceeded) it is history and has no affect on future rolls, including the roll you get from barbs.
To illustrate this if you need an 11 your chance of success is 50%. If you roll 2 dice and take the lowest the chance of success is 25%. In this case it is a +5 like you said.
HOWEVER if you roll 1 dice and that dice is already 11 or above, the chance of rolling below an 11 with the second dice is 50%, the exact same as before the first die was rolled.
Now if the spell were different and if you cast the spell before the first roll and it gave the roller disadvantage, then it would be like you said. It would actually be a more powerful against an easy to hit number if this was the case.
I am comparing "PC with Barbs" to "PC without Barbs", not "Use Barbs" to "Not use Barbs".
A PC with barbs lands spells 75% of the time that a PC without landed 50% of the time at a cost of 0.5 level 1 slots and reactions/cast.
When you cast and decide if you want to use Barbs, that PC has a 50% chance to turn the successful save into a failure.
The two numbers - 75% and 50% - are both true, because they are measuring different things.
I agree on the crits, not on the legendaries. Most monsters with legendary resistances usually also have high saves, making this less useful when the monster has already suceeded.
No, you use it when the monster succeeds.
Suppose you find a weak save and a spell the monster won't want to land. As an example, synaptic static and an int save.
The monster has a 2/3 chance to save. So you throw out static; 1/3 of the time a LR is used, and you are done. 2/3 of the time, you barb and force a reroll; again, 1/3 of that leads to a LR use.
You go from 1 action 1 5th slot for 1/3 LR removed, to 1 action 1 5th 1/3 reaction 1/3 1st for 5/9 LR removed. That is 1.67x as many LR removed.
What more, if there is more than one spellcaster (or non spellcaster save inducer), and your barb wasn't needed, you can use it in their spells/effects.
While 5/9 LR per turn isn't a lot, 2 spellcasters are now stripping 10/9 per round instead of 2/3, and using 1st level slots instead of high level ones (how many 1st level slots produce effects a mosnter eould burn a LR on?). So around turn 3, at an average of 30/9 LR burnt, the monster starts risking using up their last LR on each effect. So you can start landing debuffs.
The same 2 spellcasters would have to go 4.5 rounds of 9 high level spells, instead of 3 rounds of 6 high level spells, without Barbs.
And if you find a weaker save than 2/3, Barbs makes LR go down even faster. A 50/50 save without Barbs takes 6 caster-rounds to beat 3LR; with Barbs it takes 4 caster-rounds.
It isn't monk stunning strike spam good at stripping LR, but it ain't bad. A monk can burn ~4 Ki/round (flurry, 3 stuns) to land 3 saves/round, but it is a con save (no ability to focus on weaknesses) and a secondary attribute based DC. At level 8 this monk can force 6 saves in 2 rounds; if the monster has a 80% save rate, this is 1.2 LR burnt/2 rounds. At 2/3 save rate, this is 2 LR burnt. At 50%, this is 3.
Assuming the Barb caster has a notch better chances (can pick stat), at 2/3 it is 10/9 LR in 2 rounds (close to monk 80%), at 50% it is 1.5 LR/2 rounds (lower than monk 2/3). So barbs brings a spellcaster close, but not quite, to monk ability to strip LR.