I think the issue is that Google stopped supporting Java and NPAPI plugins. Java was scheduled to stop working sometime during April. The latest Chrome update must have finally blocked all Java and NPAPI plugins. Google wants developers to use PEPPER plugins instead. Use MyTrezor.com in Firefox or install Electrom 2.0.4. Both are still working with Trezor.
Yeah, until they remove it completely, you can still got to chrome://flags/ and Enable NPAPI.
Thanks! That fixed it in Chrome for me, and it seems to work well now. But as you mentioned, I expect that this option will be removed fairly soon so I should probably migrate to Electrum or something similar for using my Trezor.