dave2008
Legend
Action surge is once/twice per short rest, not per day. Depending on the day, that would be between one to six uses in a day.
That said, a level 20 character would make an Olympic champion look like a complete chump. By time you reach 20, you've pretty much achieved demigod status. For example, a level 20 human monk with the mobile feat can dash at a speed of 38.4km/h while also being able to casually carry a minimum of 120lbs on his back. That's faster than Usain Bolt... while also carrying at least 120lbs on his back. And unlike Bolt, who gives his all for that 9 second dash of his, our Monk friend would have no trouble doing his little stunt 20 times as needed with only an hour needed to recover (while also fighting inbetween). That's insane physical capability by any normal standard.
In comparison, a fighter would generally be expected to pursue a similar degree of martial skill to a monk's pursuit of physical prowess. If our level 20 Monk can make Usain Bolt look like a complete amateur, what does a similarly accomplished Fighter achieve compared to any Olympic level fencer, let alone a casual practitioner? Keep in mind, by level 20, any martial class has achieved a level of skill and physical prowess that puts them on par with mages who can literally bend the very fabric of reality. 8 Attacks with any weapon over the course of 6 seconds isn't exactly unreasonable at that point.
If you want to try to be reasonable about things, physically inclined characters probably begin to far surpass the whole "Olympic champion level" of things anywhere between levels 1-5, depending on what you intend to measure.
Do keep in mind, if you still want to keep with the whole "fewer attacks being made" bit without worrying about the class balancing act, you could always just fluff any multiple attacks made against a single target to simply be a single excessively strong or decisive blow. Each attack roll doesn't necessarily have to be a measure for an individual blow, it could instead be a measure of severity for a single one. Hit some poor bastard with an action-surge-pumped 8 attack rolls and the damage rolls were enough to thoroughly obliterate him? Turns out it was actually a single perfect strike right through his heart. On the other hand, if your player does want to move around and split some attacks up, he then always has the option to do so, if need be. At the end of the day, DnD is a game of imagination; and as much as I love to metagame and number crunch (and I really, really love to do that), there are situations where focusing on imaginative solutions over mechanical ones can offer some much more intuitive results. And I feel this case is definitely one of them.
To clarify I am not overly concerned with realism, and I agree that level 20 characters are demigods. I have a bigger issue with multiple attack rolls. Less so the number of attacks. An option might be to just roll the dice once and apply the result to each attack. Or once for every 2 attacks or similar. That being said, I like the feel of a fighter that does more damage per hit. Its more of a feel/aesthetic thing than a realism one.