I'm able to play a number of non-C2 HTML5 games on the PS4's browser with no problems. It recognizes controller input and everything. The only hitch is that the games I've tried won't play sound. Some games are a bit slow, but most I've tried run fine including a fast racing game called Speed Club Nitro and a shooter called Cartoon Flight which plays almost perfectly except for the lack of sound.
With C2, some games from the Arcade run okay on PS4, some slow, some not at all. Some allow for PS4 controller input like the Portal Physics Balls demo, while others are built specifically to be played on PC so they don't respond to the PS4 controller at all. None have sound.
With sound on the PS4 browser, I have however noticed that some non-C2 HTML5 games will briefly play sound but it dies in a few seconds.
Considering that there are a number of HTML5 games that work perfectly (minus sound of course), I think there is a lot of potential for HTML5 on PS4. One thing Sony could do to help (but I'm sure they won't) is give users a browser option to activate the PS4 touchpad and also allow users to change the cursor movement speed for the PS4's browser, that would be a huge help as some games like Nut Run 2 (a non-C2 game) require you to tap different parts of the screen quickly which the controller doesn't make a simple feat.