Cannonball I found a solution that works--for whatever reason, if you close the app in windowed mode, the bug happens. But if you close it first while using the special viewer i.e. the kind that you get when the site supports DeoVR's special menu (like you get at deovr.com itself), then the next time you boot up the app everything will work. This fix works even if you boot it up "broken" once, navigate to deovr.com, put it in tile mode, then close the app.
It still happens for me (on literally every website, I don't know why they keep asking, it's definitely an issue with the app itself and not any particular html/css), but at least this workaround means I can use the browser.