I’ll address the claims of water resistance first. Any time a device is opened (phone, watch, tablet), in most cases, yes, this will affect the water resistance of the device. I’ve never heard of claims that these devices are recommended to be replaced after the seal is broken. Sounds like something Apple wants consumers to think so they can sell you a new watch. Of course, the newer watches are designed so that you can swim with them. In this case, I can see the logic. I definitely would not recommend swimming with the watch after a screen replacement. The first generation to my knowledge was not designed for this anyway.
With that said, I have seen some watches come back to our shop because the adhesive didn’t take very well. This might be more of a problem with the force touch sensor coming loose however. The screen sits on top of the sensor and our thought is that if the sensor is loose, the screen won’t stay on very well.
Second, you’re completely correct. There is no easy way to get new adhesive on the device without taking the watch apart. This is one giant pain in the rear especially since Apple had to design the worst system for how the screen attaches. If you look at the guides, they place a bracket over the connectors. I kid you not, I spent about 20 minutes just the other day trying to remove the clip and I’m a pretty seasoned tech. I can only imagine what it would be like for a novice.
If this watch come into our store it’s something we would repair but honestly, with it being a series 1, my recommendation would be to honestly purchase a newer watch. The battery is likely not great and needs to be replaced and once you factor that in plus it’s age, it’s probably worth the upgrade.
Also, just a note about Apple watches. If you break the screen, it’s almost worth just purchasing a new watch. In most cases, the repairs cost about as much as a new watch. If you find a shop that has a low ball price, I would question it.