I think it all boils down to playing style. If you have a more aggressive approach, then you're more likely to use Reckless. Other wise, Regenerator.
Both have pros and cons, which I'll try to give an unbiased version of:
Reckless is suited better, I feel to more experienced players. Given that Mienshao has practically one or two moves that make use of said ability, by choosing it, you are effectively using Mienshao for its basic role of a glass cannon. These sets ate generally Life Orb, and are for the sole purpose of using brute force to overpower defensive cores, making 3HKOes into crucial 2HKOes, and 2HKOes into OHKOes.This however, requires a good deal of patience and understanding of how RNG (hax) works, and having a back up in case High Jump Kick misses. Utilizing Mienshao for the sole purpose makes a good amount of sense, because the extra power might make or break the game in higher tier and higher ranked battles. It recognizes the frailty of Mienshao's poor HP and substandard defenses and goes straight for the KO, also taking Life Orb damage into account.
Regenerator, on the hand, is suited for newer and more conservative players. The Fake Out + High Jump Kick + U-turn combo is a great strategy that seems too good to pass out. Fake Out, hit hard once, or regain health even if you miss, and then use Fake Out again (after coming back since you switched out by U-turn). Decent idea, on paper. It works well with weaker teams, where you are using Mienshao in a more defensive manner as a pivot rather than an all out attacker. You are probably scouting, or breaking Focus Sash, or generally weakening slower opponents, as opposed to using extra power to overwhelm them. Life Orb damage is also covered by U-turn + Regenerator, so you are comfortable about the HP loss.
Thus, you can argue that reckless players utilize the high risk- high reward idea that might end up sacking a frail Pokemon like Mienshao in exchange for winning the game, much like losing a battle to win the war. Knowing what to sack, and when to, is an important tool to winning, and it is much easier to sack Pokemon that are low in HP, as opposed to sending out Pokemon with 2/3rd or more health. On the other hand, Regen users tend to preserve Mienshao to use it to a greater extent in the endgame, thanks to its good speed.
As you can see, both have their roles, and your playing style and agenda for the battle that makes one better than the other. It is thus based on other factors like the tier you are playing on, other Pokemon in your team and their roles, your opponent's team, skill level of both battlers, and so on.