On the contrary, it's the clear cut endgames where there is most information given by a rejected no progress claim. If the players already know that there is no chance of a win by the side claiming, as e.g. in a claim by Black in a KBNK position, then a rejected draw claim confirms to both sides that White has has a theoretical win.
But there is less to be deduced if the ending is not clear cut. Suppose a few moves after the "monster" position you gave White promoted his pawn and reached, say, this position.
I think you're being deliberately obtuse here. Unless you're a complete beginner, you know that the endgame KBN v K is a win 'always', so if the arbiter were to reject the claim after 50 moves, you can quite confident that you're actually quite close. Never mind, I used it for illustrative purposes, because the result can never stray beyond draw for the defending player.
The monster position was just another example, asking where on earth you're supposed to draw the line. The man behind the bar is going to see that it's still a win, so why accept the draw claim? With 500 moves between the last pawn move and the first capture, I dare say there's plenty of opportunities to stray off the most correct line of play (because it's so complex), but where do you the draw line? When should you no longer get an extension? The barman wouldn't have a clue.
... In that instance the draw period would be 25 moves and the game should only get to 50 moves if Black claims first (which, come to think of it, is why I originally made the claim period 50n+1 ply) at which point if Black has not progressed since he claimed a claim by White would succeed. ...
On reflection it would be much better to keep the draw period at 50n ply (25n moves) and just let the players claim a draw on either player's move (possibly simultaneously). KRK and KQK endings in which the stronger side doesn't "progress" would then be limited to 25 moves on a claim by the weaker side.