The Champion's crit chance is a really bad damage feature, if your goal is to compare it to the effectiveness of other features. If the goal is simplicity über alles, then it is of course great because it's one of the simplest possible damage sources in the game.
Let's say we're looking at a Champion Fighter with 15% crit chance (crit on 18, 19, or 20), a typical net hit rate of 60% (this may be slightly low or slightly high, but 60% is a good ballpark; crits are part of this value) and using a greatsword (greataxe has bigger highs, but lower lows and the numbers are better for greatsword unless you're a half-orc). We'll further assume great weapon style since the Fighter can easily have that. Feats are irrelevant in this context (they're unaffected by crits; the Champion gets the same bonus from them as anyone else.) 20 Strength, of course, since any Fighter should have that by this time.
Average damage per regular hit: 4.125+5 = 9.125. Average damage per crit: 2*4.125+5 = 13.25. Expected damage per attack: .4*0+.45*9.125+.15*13.25 = 6.09375. Then, by using the expected damage without this crit benefit (.4*0+.55*9.125+.05*13.25 = 5.68125) we can calculate the average damage gained per attack. While numbers at this fine a degree are not useful for singular attacks, we're looking at averages over entire adventuring days, perhaps even over several of them, so it will be a good approximation. The Champion Fighter gets (approximately) 6.09375-5.68125 = 0.4125 extra damage per attack (averaged across all attacks, not just all hits.)
For the sake of argument, we'll take the most favorable, designer-expected conditions, that is 8 fights every day and three short rests so this Fighter is getting maximal use of Action Surge in combat. We'll also assume that the Fighter is never unable to attack anything (perhaps the axe can be thrown, but magically returns to the user, I dunno.) A typical fight is not longer than 4 rounds, usually less. That gives us 4*8 (regular combat rounds) + 4 Action Surges (won't get more per rest until level 17) = 36 combat rounds. Each combat round is 3 attacks at this level. So the expected damage increase, under these conditions, is 0.4125*36*3 = 44.55 bonus damage.
By comparison, a Battle Master that uses her maneuvers, whatever they may be, has 6d10 per short rest. I will assume inefficiency, thus this Battle Master always holds onto at least one die at the end of combat, thus never gaining the benefit of the "if you have no superiority dice and roll initiative, you regain one superiority die" feature. With four rests (finishing the long rest and taking 3 short rests), the Battle Master gets a total of 6*4 = 24d10 dice. Because there are several maneuvers which only trigger when you've hit the target and which just let you add the superiority die to your damage, there's no need to perform any hit calculations, so long as we assume that the aforementioned Battle Master can manage to land 6 attacks before each rest (an extremely safe assumption). You just get a bonus 24d10 = 24*5.5 = 132 bonus damage.
The Battle Master gets at least triple the extra damage that the Champion gets, under circumstances taken to be almost maximally favorable. IOW, Superior Critical is garbage. Even if you cut out one of those short rests, costing the Battle Master 6 superiority dice, 18*5.5 = 99, still more than double the Champion's bonus damage (before factoring in that the Champion just lost 3 rounds of crit-fishing due to getting one less Action Surge!) Further, that's in extremely unfavorable circumstances PLUS the Battle Master literally never triggering their minor recharge. (In general, it's also more efficient for them to instead take Great Weapon Master as a feat, if allowed, and use Precision Attack to mitigate the loss, since that always gives +10 damage but becomes an average of 1d10-5 = +0.5 to hit, meaning you do slightly better than breaking even on your hit chances. This is one of the reasons some DMs dislike such feats.)
Edit: Now, I didn't factor in Advantage, which would help fix some of the impact. But even with how profligate 5e is with advantage, I don't think you're going to get it often enough to have that much of an impact.
With advantage, your crit chance becomes 0.2775 while your hit chance (presuming the aforementioned 60% normal hit chance) becomes 1-0.4*0.4-0.2775 = 0.5625, vs. the normal 0.0975 crit vs 1-.4*.4-0.0975 = 0.7425, giving expected Champion damage-per-attack of .5625*9.125+.2775*13.25 = about 8.81, while non-Champions would have 0.7425*9.125+0.0975*13.25 = 8.07. This gives a net damage gain of 0.7425 per attack, or 0.7425*36*3 = 80.19 extra damage. Which is definitely an improvement, but it's not even 2/3 the bonus that the intentionally-inefficient Battle Master can bring, if the Champion is somehow always getting advantage on all attacks.
