My Apple watch series 1 screen is coming off due to the battery swelling. Is this a defect?

I don't believe it is a defect. It may happen with age. To understand Apple Watch performance and its relation to your battery, see:

Apple Watch - Battery - Apple

Apple Watch Battery and Performance - Apple Support

“Battery life” is the amount of time your device runs before it needs to be recharged. “Battery lifespan” is the amount of time your battery lasts until it needs to be replaced. Maximize both and you’ll get the most out of your Apple devices, no matter which ones you own. See the "Tips for Apple Watch" section in:

Batteries - Maximizing Performance - Apple

For repair & service:

watch Repair - Official Apple Support

watch Service Pricing - Apple Support

Haha! Is it a feature? No! Of course it's a defect.

Whether it is a design defect or a manufacturing defect doesn't change anything for us users: we bought a watch that is not supposed to loose its screen. Period.

I replaced a bloated battery twice on my wife's Series 0 due to the screen popping out. I thought the issue with hers was that she mostly left it on charger and didn't wear it much. So both times I bought an original battery and followed a YouTube tutorial to swap it out properly (both times easily done), but they both only lasted around a year before puffing up and popping the screen out again.

But now I just noticed my Series 1's screen has become loose in the front (side facing me), and I can tell by the bounciness of it that the battery underneath is definitely applying some pressure.

Don't get me wrong, I love my iPhone and Apple Watch. The adhesive holding the watch screen was just not a good design choice. I don't think it was deliberate, but I would expect a serious manufacturer to own their mistake and properly compensate customers. In my case, I would favorably consider an exchange for a Series that doesn't have the issue (is there one?) or a full refund (even towards the purchase of a Series 6 for example) or a free battery swap by Apple every time it pops out. Anything less will feel to me like a cheap trick to get me to buy a replacement.