Found a fix, working for me now.
I use Brave browser which has heavy shields and tracking blocking features (other browsers like chrome, safari also have similar blockers).
I disabled the trackers and blockers for the Figma website. (Present beside the URL bar for each website when opened)
Reloaded the web page and I got the Open in Desktop App option
Non-mandatory step: re-install the desktop app post steps 1 and 2.
None of this worked for me and I try to prevent sites and apps from tracking me as much as possible. I wrote a simple bookmarklet to open links in the Figma app.
To use this, make a bookmark in your browser for any site—I’d suggest having it as whatever your browser considers a Favorite so it’s easily accessible. Go and edit the link and paste in the following javascript:
javascript:(function() {
var oldUrl = window.location.href;
var newUrl = oldUrl.replace(/^https:\/\/www\.figma\.com\//, 'figma://');
window.location.href = newUrl;
})();
This works on macOS Sonoma. I have no clue how Windows works, but perhaps this works here too.