Hey, I know this post is ancient, but did this ever change for you? Or did you ever find anything out? I've been having the same issue on my machine (latest Steam version on Arch) for quite some time, but I always just re-started Steam to "fix" it. However, it's starting to annoy me more and more, so I'm trying to finally get rid of this issue. Did you maybe fix it somehow? Or anyone else who sees this?

To anyone with the same issue reading this:
Check if you're using joycond (and uninstall it). After trying to find more reports of this issue, I did not find much, so I tried going to the most basic setup (i.e. removing joycond). That simply resolved the issue; Steam now just immedeatly detects my controller as I connect it. My guess is that the issue arises due to joycond "delaying" the controller from being ready (so Steam tries to use a newly created controller, but since joycond waits for another user input, that fails, and when joycond is ready, Steam won't notice? idk if that's what is going on, but it'd kinda make sense, especially since it works when starting steam after connecting the controller, so that when Steam checks all controllers, the controller is not in joycond-setup).
Anyways, while it does indeed fix the issue for me (I don't really need joycond, I originally set it up when I only had a pair of joy-cons which I wanted to be treated as a singe controller device), I still think that it'd be better if this was actually properly fixed, since using joycond is not that uncommon.