Since you have confirmed that the browsers are supporting HTML5, I would suggest we also check for proxy settings if any. Also for reference please see the KB article below with a number of questions around supported browsers around the issue:
Have you installed VMRC on the Windows 10 box? For me there are no problems under Windows 10 (32bit and 64bit). I found it even working on Windows server 2016 (if you install flash player).
Chrome and Edge I havn't tested yet for this pupose, but FF (for the Flash version only ESR 45.9.0 is usable, the later versions are awfully slow) does work, too.
So it is probably a local setting, which is interfering.
I finally got the Flash client working in Chrome. Even though I had set "Flash=Ask First" in chrome://settings/content , it seemed to start working after clicked "Secure" beside the URL (when I loaded up the page), and explicitly set Flash=Allow. It was like it wasn't even bothering to display a prompt before.
I also recently did another Windows 10 update to 1703.15063.674.