Why Did Fischer play 17) ...Qxf4

No, he wouldn't.

He would either play

1) 18. ... Qb4+ 19. Bd2 Qe4 (when White has both a Queen and a Rook under attack) 20. f3 Qh4+

... or the same moves in the other order:

2) 18. ... Qe4 19. f3 Qh4+ 20. Bf2 Qb4+

I know. Fischer missed the win before the sealed move. Botvinnik gave 43...a5! with a probably won rook + pawn/pawn ending.

Wish Botvinnik and Fischer had played more than once when Botvinnik was at his peak. I think Fischer was only 19 when he played Botvinnik in 1962. To draw against the world champion at that age is pretty impressive. I read that Fischer thought he had a win if he had played a different move in the endgame but Botvinnik may have disputed that. Young Garik (as he was known back then) Kasparov analyzed another way Botvinnik could have drawn if Fischer had played differently.

But my memory's not the best as you can tell from the beginning of this thread:)

Was a really interesting game. It looked like Botvinnik was going to steamroll Fischer with those advanced central pawns in the opening

Indeed, Fischer and Botvinnik both reached an analized position in that game as (in view of Black after 64 h4=Q):

I shortened the analysis a little bit. It seems that Botvinnik and Kasparov were right here.

Kasparov hadn't been born yet in 1962, so I seriously doubt he was known as 'Young Garik' or that he analysed a draw at that point...

The game was analyzed well after 1962 and Botvinnik and Fischer traded analysis (though not directly.) I'll try to find where I read that, though

Kasparov's analysis of the endgame (in which he supports Botvinnik's claim that it was a draw even if Fischer had played differently) is cited in a few places on the Internet. Kasparov was a school boy at the time (one account gives his age as 13, which would have put the analysis in the mid-70s.)

I read it in either Frank Brady's or Andy Soltis' book on Fischer, neither of which survived the purge of my chess library about five years ago

Someone on the Internet quoted the source - it was Soltis' book, "Bobby Fischer Rediscovered."

"Moreover, Botvinnik said that one of his students, a youngster from Baku named Garik Kasparov had found a second draw - 67 Rc4 Rb3+ 68 Rc3! Despite many checks. Black cannot make progress (68 ...Qe1+ 69 Kd3 Qf1+ 70 Kd2 Qxf2+ 71 Kd3). Botvinnik hinted that his opponent had been out-analysed by a 13 year old schoolboy. Fischer never replied - and this is almost certainly the closest we will ever get to a Fischer-Kasparov match." (Bobby Fischer Rediscovered, Andy Soltis (2003))

Yes - the point is that the way you said it ("at the time") implies that it was 'at (or shortly after)the time the game was played', as you didn't reference any other time.

This is not the only source! Botvinnik wrote a book about the Grünfeld, where he analized this game and mentioned the "other way of draw" found by his 13 years old pupil Garik Kasparov (which dates this analysis after April 1976).

No WanderingWinder, when I remember Psalm25 correctly he wrote: "later young Garik Kasparov found an other way of draw."

the first move order doesn't work. 21.Qf2 is possible.

The lucky thing is that we don't have to rely on your memory, because we can actually see what was written, and indeed it looks like you aren't remembering him correctly, as the word 'later' doesn't appear in his post:

When Psalm25 wrote 'Young Garik Kasparov' you should be aware that this analysis must have been 'later' because G. Kasparov was born in April 1963.