Does the typical game being a game which plays in browser, rather than a game which requires a download, make you want to play it more?
Not really. I don't typically try browser games that I don't like the premise of. I wouldn't want to play a browser game just because it's a browser game.
If it helps, I tend to look at premise first - how it looks, feels, and what it's about. If there's a browser version available for something that's caught my attention, I'll play it there on the fly - it makes it easy for me to just scroll down and write a comment for it. If the game requires a download, I probably won't play it for the time being until I can either find time to sit down and install it, or play other bite-sized browser games with a similar premise first before coming back to it... if I remember to, that is. 😅 If I'm on my phone, I will only play browser games.
Of course, the download-required games would need to be from publishers I know and have an amount of trust in, such as particular developers I've chatted with (either from the same writers' group or game jam), folk from the same fandom or a reputed publisher like Y Press Games. Large, long, and graphics-heavy games tend to be better downloaded, but for safety reasons I do try to ascertain that the developers are genuine first before downloading them.
I feel browser versions of games make them more easily accessible on mobile devices or people who're casually browsing for something quick to play, which in turn makes them look more desirable as a whole statistically. Ease of access can be quite a draw, but I personally find first impressions - thoughtful synopses, nice cover images and screenshots, attractive character profiles - to be more successful in getting me to try them out overall.