The "allow-host-input-processing": false should have no impact on any Steam features other than the overlay. All other features like achievements should still work.
If you add "allow-host-input-processing": false I'm pretty sure that you don't see the Steam pop-up "press SHIFT+TAB to show the overlay" (or something like that), so I'm assuming that you will not see the pop-up for achievements as well. The achievement will be probably counted but I don't know if the player will be notified.
EDIT: I was wrong, the Steam pop-up works even with the line "allow-host-input-processing": false... Curiously, Discord's overlay works without any problems.