Yes, the display would still be usable without the adhesive, the clips and the security screws at the bottom of the phone are sufficient at holding it in place.
As far as waterproofing though, that’s going to be the tricky bit. Providing water resistance is the primary function of the adhesive liner. You can still try to make use of some more generic adhesive like this Tesa tape, but it is definitely not the same level of water resistant as Apple’s original stuff and it will be difficult to put it in without leaving any gaps for liquid to enter the phone.
Tesa 61395 Tape
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