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Simpler does not necessarily mean less drawish. The one extra move could hardly be a serious concern, as in practice people would resign when mate or capture of their King is unavoidable.  It seems that your real gripe with FIDE rules has nothing to do with checkmate, but with the fact that stalemate is a draw.

Have you any idea which fraction of FIDE GM games end in stalemate? Declaring stalemate a loss will have almost no impact on the draw rate. Have you measured whether the fraction of draws in engine self-play is measurably lower when you let them play under rules where stalemate is a loss?

Your idea of making the King an immobile piece is interesting. (Whether such a game would still qualify as 'Chess like' is debatable, of course.) Your claim that this would decrease draw rate seems unproven, however. My guess is that it won't. Players will just reserve more other material to evade checks, when the 'King' itself (I guess 'Flag' would be a more appropriate name for such a piece) cannot step out of check. Material they could no longer use to attack the opponent, as making it leave their defensive positions is suicidal. What matters is the number of ways you have to evade check. Whether some of these ways are by moving the royal piece or not doesn't really matter.

E.g. consider a 'Chess variant' with initial position: Flags on a1/h8 Rooks on b1/g8, Pawns on a2/h7. Seems a dead draw to me, you can do nothing but moving the Rooks over the back-rank. with an occasional excursion to protect the Pawn from the side when the opponent attacks it.