You're right that the scriptability of Tiled needs more visibility. However, I am not sure that a script browser is the way to go about it, because unlike browser extensions, most people who write scripts do it for their own personal or their team's use, to accomplish highly specific tasks that only they need, in a way that they specifically prefer.
Scripts that are useful to many people are better off being written as Tiled features instead. Scripts that are general-purpose but still useful only to a small minority of users are the scripts that benefit most from a script browser, but they're also annoying to browse no matter what, because they're not going to be relevant to most people.
I think a script manager would help with the visibility of the scripting feature and with most of its inconveniences without all the downsides of a script browser.
Then again, I'm also someone that learns about Firefox extensions from hearing people talk about them online and following links, I never use the Firefox add-on browser xP
I forgot another disadvantage to a big centralised repo: Issue tracking. It's much easier to get and provide support when a repo contains at most a few scripts. Reporting issues in one script when a repo contains dozens or hundreds is quite annoying, especially if there are multiple scripts that do similar things or have similar names, but even if they're all completely distinct. Even worse is responding to these issues, because script maintainers will not have a quick way to see just those issues that pertain to their work.
Lastly, I also worry that script developers will be even less likely to submit to tiled-extensions if doing so means people will randomly find their scripts, because it puts more expectations of supporting those scripts (most people just want to write a script, maybe put it out there, and forget about it), and because the manager will make people less likely to actually look at the script, which means they'll miss potentially crucial information that would normally be in the script's readme in its repo, or at least in the comments in the script. This second problem could potentially be helped by showing the first comment in the script in the browser when you click to view a script's details, though.
I've written a bunch of Tiled scripts that I'm happy to see other people use, but I don't think I'd submit them to be available in this sort of repo and script browser, because it would just be too stressful for me.
Edit: A script manager could also have a link to the tiled-extensions repo (which in turn could continue to link other repos), too.