I don't think the amount of draws or stalemates has much of an effect on how chess is perceived by the general public. I've often heard people say that that's the way it is, but I've never seen any data to suggest that this is actually the case.
The partial scoring system that was suggested here is absolutely bonkers and would make watching chess games incredibly tedious. A stalemate is NOT between a draw and a win, it's a draw. The game has basically crashed, there's no continuation and there's no mate.
IF something like .7 points for stalemate was introduced we would end up having an endless number of unnecessarily long, long, long games. An endgame with multiple pawns on the board, but where one side has a very slight, but not decisive, advantage would be played out when both players would normally accept a draw. There are so many endgames where a slight advantage is not enough for the win, but with stalemate a preferable over a draw the side with the tiny advantage will just keep going forever.
Personally, I don't see why a stalemate should be worth more than any other draw. Stalemates usually happen in situations that are dead drawn, usually the king +p vs king. Any rule change seems like it would just be to appease the beginners who still stalemate when mating with the queen and rook. The impressive manouvers where the player who is in the worse position sacrifices his remaining pieces to achieve a stalemate are very rare, but they are definitely impressive enough to be worth that half point.
If we are going to reward drawn games differently then I would rather see threefold repetitions rewarded over stalemates. Being able to force a repetition is often much more impressive than a stalemate, which is the inevitable result of many drawn endgames. Of course, I don't advocate scoring any draws differently, but this way makes as much sense as any other proposal.
Raging B: Well, yeah, if you see the high percentage of draws at the GM level as the big problem, the best solution by far is to switch to Chess 960 or something roughly equivalent. It's not a very popular change with club players, but the GMs seem to be in favor of it. Chess 960 would certainly eliminate the deep opening preparation that is killing chess at the top level. Take out the memorization and go back to playing chess.