Deserves got nothing to do with it.
I think the Heat circa the streak were the best team in the league, but peaking during a point in the regular season doesn't get you the title. Wade, Battier, and I think LeBron are worse now than they were then, and not just because of how the Spurs have been playing them. When you factor in last year's compressed schedule and title run, the Olympics eating into any offseason recovery, and another run to the finals against some really physical teams, it's not surprising that they're really hurting or fatigued.
And yeah, "nobody's 100% at this point in the season" but there's a difference between carrying a bit more soreness and, for example, what Steph Curry was dealing with after the first game of the GSW/SAS series. Pop's management of the team's minutes (and the extra burn it gave for bench players who might have to occasionally take on bigger roles) during the regular season has probably benefited the team in the Finals as much as any decisions he's made during the series itself.