Firstly, I feel we should distinguish between that which qualifies as "not normal means" and that which doesn't. If a player makes a single illegal move, it would be overbearing to assume this was part of not normal means to try to win - in this situation especially, an exchange down is not a huge material deficit in the first place. If the arbiter is playing over only the last 10 moves, isn't that a bit short-sighted? I agree it's a possible scenario but we should assume the arbiter is competent and should play through the game from the point at which material was gained, or from the start. The arbiter would also, I hope, give benefit of the doubt to the plaintiff re the illegal move.
This all leads to another point - if as Michael says, FIDE rules are being strictly interpreted, and the game is in progress, then the arbiter/players can re-convene, add 2 minutes to A's clock, and the game be supervised, with the arbiter to declare final result. If the arbiter is in no reasonable doubt about the correct answer, this is irrelevant. When adjudicating, the arbiter can look at the whole game, but pay particular attention to manoeuvres or moves designed to gain B some thinking time, or rather, deprive it of A. A similar example, to my mind at least, would be in a K+P v K ending which is theoretically drawn. If a player repeated even one set of moves as the aggressor against a low-time rated opponent, I would be seriously inclined to award the draw if it were claimed, since a repetition shows implicit lack of desire to make progress over the board.
My own personal feeling is that an illegal move without adequate supervision should be treated as accepted by both players, just as a result is once the scoresheet is, as here, signed, since it should be claimed as soon as it is made. Imagine the controversy if the arbiter adjudicated the game from 66, and that before that B made no attempt to win solely on time, and even had a winning position, lost because of his illegal move. This would be close to bringing the game into disrepute I think.
As far as positions, if both sides have any decent portion of material A would be hard pressed to have his claim accepted without particular reason. The closer you get to an endgame without glaringly countered pawn structures or anything odd, the easier it gets for him, and of course pure R V N/B can be given the draw unless there is any danger to A's King.
If venues close at a fixed time, the easiest way is to play with a margin for that closing time, and start earlier, which sounds very haughty but is very true. Up here in Cumbria all of our clubs have digital clocks bar one or two; but I doubt any of our clubs are yet solely digital in matchplay (Carlisle, my club, certainly isn't, though I hope to change that in future), so we use a straight G90 for city championship games, and G120 for county league matches. There is a clause saying Fischer time +30/move can be used with mutual consent, but to my knowledge it's never yet been invoked. Perhaps I'll have to be the first - I have to get my time trouble under control somehow!