Well let's do some math, he ended his career 696 hits short. In his age 33-35 seasons he hit .307. Say he regressed down to .300 after that, if he got the average amount of AB's each season after that based on those years (which is 533), he'd average 160 hits a season. 696/160 = 4.35 seasons. Considering he averaged 134 games played in those seasons (before that he never played less than 146 except his rookie season), I'd wager that's a conservative estimate. He was still a great hitter when his career was abruptly stopped, so yes, I'd say he easily got there considering players routinely played to age 40 and beyond in that era. Hell, Puckett was a better career hitter than Paul Molitor, and Molitor posted .330+ averages more often than not after age 35. By OPS, Puckett's age 35 season was his 5th best ever and trailed his top mark by only .026 points. He wasn't turning into some scrub as a hitter like you seem to be suggesting. Was he a great fielder anymore? No, but that wasn't affecting his batting.
On the left, you see Kirby Puckett in his age 35 season. On the right, you see Paul Molitor in his age 35 season.
It's not hard to see why one doesn't believe that Puckett was going to age gracefully into his age 40 season and why Molitor did just that.
Oh, and Puckett wasn't a better hitter than Molitor. Their career OPS+ are 122 and 124. They were basically the same hitter (though as an overall player, Molitor was far more valuable over the course of his career than Puckett, sporting a WAR of 72.5 to Puckett's 48.5).