This is an easy call for me.
Consider your your stack vs the blinds. You're sitting on 10BBs with high blinds. Even without the antes factored in, you're losing over 14% of your stack each blind rotation. So, you really cannot wait for quality hands. What if you're down to 4K, you get AA, and you win a pot to bring you up to 10K again. Well, it didn't do you much good because you're still at 10K and blinds are probably higher, making you situation even worse. So, you're looking for opportunities to get your money in and potentially double up. Especially when you've got a very strong starting hand and your stack vs blind is still potentially strong. Would you fold here if you only had 5 BBs? Hopefully not, but how much more is a 10BB stack going to be (assuming you won and doubled up) since you'll still be short stack at 10BBs. At least if you double up now, you're sitting on over 20BBs, giving you a lot more room to maneuver and increasing your chances at winning.
Consider villains shoving range. You say villain was a tight player, but it could be the fact that they're also nearing the short-stack so their shoving range is much bigger (as it should be). If they've shoved twice PF in the last 100 hands, then I would say yeah...they're pretty tight and I can see a fold being a good choice here. But if they're actively shoving to steal, well then AK is a great hand to call them with. In most cases, you're crushing their shoving range with AK. You might be up against an under pair, but I'd be happy to race at this point. Worst case is you're up against AA or KK. Most of the time, you beat villains shove range. If you're not calling here with AK, what hands are you calling with? What hands are you shoving with?
Consider what you goal is for the tournament. If you're happy with winning a $1 and some change for your hours of effort, then fold here and make it in the money. However, if your goal is to win then you're going to have to take these chances at this point in the tournament. Just because you're 4 from the money should not alter your decisions if you're looking for a 1st place finish. Would your decision change in this hand if you're 44 places from the money? It probably would because you've still got a ways to go if you want to win, so why does that change when you're 40 spots closer to winning?
Consider that its a freeroll and you have nothing to lose but so much to potentially gain. True, buy-ins or lack of should not be an influence in your decision making, but it can be in this situation. Well, actually, you're losing potential profits you're giving up by passing opportunities like these. You say you finished 20th for some nice change. Lets say $5. But if you had played an won this hand, you might've finished higher. Lets say you made FT and finished 8th for $22. Well, you've just given up $17 in potential profits. Of course, we'll never know if you would've won or lost but the point is that these are the chances you (especially at the later stages of a tournament) if you're looking to finish in the top spots.
So, lets see...
-Short stack and blinds are high
-Big starting hand
-I crush SS villains shoving range
-Worse case scenario: Im in a coinflip situations, only fear AA and KK preflop
-Im near the money, but I want to win
-Perfect opportunity to double up and increase my chances of winning
-Nothing to lose, but can potentially earn a lot more in profits.
Considering everything mentioned above, I like a call here. Its a small risk for a potentially big reward.
BTW, don't get me wrong, Im not saying your fold was a bad play. Theres a lot of things to take into consideration and I dont know what would've happened if you had called. You manged to finish 20th and thats great, congrats on that.
