Dude, it's not just OBS. I'm not a streamer. Most things do not play nicely with webp.
I am a DM and I'm constantly downloading artwork to make tokens or maps to slightly tweak to fit my purposes. Webp is the bane of my existence because no one supports it. The image viewer I use doesn't support it. The file thumbnail gets stuck as a white square. The token maker website doesn't support it and the program I have claims to support it but won't load in the image. The moving webps are supposedly supported by a VTT I was considering but again they just never loaded properly or loaded as a still image. I recall sending a webp on discord and instead of displaying the image inline it just showed it as an attachment.
I don't care about file extensions. I've never had an opinion over which one is my favorite. Do I like png or jpeg? Don't care. But I hate webp because so far it has been functionally worthless to me be cause seemingly no one supports it.
Sure. Might be better quality. Might be tiny and amazing. Might be hyper efficient. But if I can't use it after I save it, or that I have to jump through hoops to use it, then he fucks the point?