I think the issue here is why would you ever give anyone but the character who can deal the most damage your extra attack? In other words if the damage were to be set at the same amount for all choices it becomes an interesting decision around tactics... but if it's 1d8 vs. 2d12, IMO...it's almost a no brainer outside of specific circumstances.
Well, we played with this a LOT in our 4e games, and it just didn't work out that way. For one thing it wasn't always one guy that did the best attack. It was at least situational. It was also quite common for one guy to be in the right place to deliver the RIGHT attack, which didn't have to be the highest damage one.
For instance the dwarf fighter pretty much outdamaged everyone else in the early going of our first game, so it would make sense to give him the attack, pretty often, but the rogue was of course pretty much on par with him, better if she had SA, but that might not apply (especially off-turn, remember you aren't in total control here of what the situation is). So it was always a choice between those two at the very least. The STR cleric sometimes was a good choice too, her MBA wasn't shabby and might just do the trick.
Also you might have had various possible riders on MBA/RBA attacks, so if you needed to push someone, maybe the answer was the guy with the force weapon, if it was undead, the guy with the radiant damage rider, etc.
Honestly, I think it wasn't an issue, at least I didn't see it as one. Now, maybe 5e is so skewed to one character having all the DPR that it just never makes sense, but IME even the battlemaster, tough as he is, doesn't outmatch the EK or the rogue consistently in that department. I THINK we can make it work.