Actually, if one has particular material that can checkmate a lone king (for example: anything except a single knight or bishop), one CAN force one's opponent to take the material or allow stalemate.
A queen is obvious: chase the king down to one end or side of the board, then place the queen on the square "in front of" the king. For a rook or bishop, chase the king into a corner with one's king present as well, and check him, a bishop for example, picture lone king on a1, one's own king on b3, and the bishop checks the king which must move to b1 whereupon moving the bishop to b2 forces stalemate. A rook is similar, but one might as well mate with the move. Finally, a pawn: eventually one will have one's own king on, say b4, pawn on b3, enemy (he's not your opponent if you were stupid enough to buy a flight that leaves during the tournament) king on b1. Moving the pawn to b2 forces either a capture or the allowance of promotion to a queen and then loss of the queen as before.
Any larger amount of material would reduce using the same techniques or easier since one could promote an extra pawn to a queen, then force the taking of all other pieces using the queen's total domination, then lose the queen as above. No pawns? Multiple pieces make the above forcings even easier.
One is not looking for a win, just the draw so it's easier as the forcings don't have to cover all moves of the other king, just vis-a-vis the piece one is losing. Lone king means no funny obstacles to hide behind (one may tuck any extra knight into a far corner if one can't keep it out of the way and handy pushing the king into a corner).
The sole impediment then is the time an opponent might have on the clock which he might try to pretend he is using to keep you from checkmating him. That would be clear unsportsmanlike conduct to anyone except a Soviet arbiter and there aren't any more of those. Combine that with a clear verbal offer of an immediate draw and there would be no question about needing the time to defend and with the opponent neatly on record as only being concerned about losing, and so needing the time, even a Soviet arbiter might not risk never being hired again.
Tedious, yes. Need the rule, yes. After all, they changed the rules specifically to not allow castling with an unmoved king and an "unmoved" promoted rook although the pawn it used to be was obviously moved so it wasn't really more than a chess problem "gotcha" so why not something useful like this? But... there's no actual impediment to pretty quickly losing the material so as a practical matter, probably not a real world difficulty. (And the airplane ticket? If you really have a flight to miss 'cause you were dumb enough to schedule a flight leaving before the tournament ended, I for one do not feel sorry for you having to resign or miss the flight. Sorry, that's not in the question itself, but it seems likely to be considered relevant by many as standing in for the class of unspecified but somehow reasonable things life might throw at us.)