On the technical side, webp support still isn't like png. Tried dragging a webp into Google Slides just now, got "unsupported image type," which is ironic. I'll try again in like 10 years.
Oh that's a good point.
I see lossless WEBP mostly as a way to save bandwith where PNG would have been used. If you've got a pipeline where, anyway, you already "crush" your PNG file, you may as well also generate a lossless WEBP file and serve that: all browsers support it. And you can fall back on the optimized PNG should the browser not support WEBP.
I mean: I use WEBP, but only lossless WEBP, as a replacement for PNG when I'd serve PNG files to browsers.
But for that one usecase: showing a PNG file in a webpage, I don't see that many downsides to lossless WEBP. It saves bandwith.
Also, if users download your images and use them elsewhere, webp will still be more annoying for them. Though it's not very common that you want them doing that anyway.
Any updated (modern) browser should be able to see webp just fine, I'd rather just serve it without a backup plan if I'm planning to have webp in my website.