Interesting news, Gallery is stopping work on the project, as said on their page people are too busy with their lives. I tried this web-app for my photo galleries in the past, in my opinion, well I didn't like it specially due to performance (a list of other issues) but this is not about that. I did play with other options over several years and some were kind of useful, most were not. There were some paid options that worked great, sorry I don't remember the names and I'm not even sure if they are still around.
I fully understand what you describe here, in full detail. The following are not perfect solutions, but it's considering the situation and options.
I use the gallery as a permanent archive of my photos so I don't want to have to change things.
You could try playing with Wordpress picture management, there are plugins yes, but I would suggest experimenting without any plugins, why? they get behind as WP upgrades versions and code gets broken. WP is not of my liking but it has evolved a lot.
Drupal (or Backdrop fork) would be my first option, yes even the old Drupal before going Symfony based. You can easily build a photo management solution and a photo gallery: no kidding. You can do so with plugins-extensions but you can also do so without them, using pure basic modules. I've worked extensively with Drupal and I know this can be done even with the basics, honest comment. Sure Drupal takes a while to get hands on (more than Wordpress) but it does work amazingly well. I decided to stay away from new versions (Symfony based) but I'm pretty sure things can be done there too. There are other Drupal expert members around here, I'm pretty sure some will share this opinion.
You are not precisely asking in this direction but perhaps it can get your interest: offline options
. I did played with a few Desktop Based Photo Album options, now old software. Several of them allowed creating diff albums, sub sections, etc, the usual but most importantly, they allowed you to create full html albums with resized pictures (to your specs, the size you want) and have them export everything on a simple package, a package you could upload to your server. This included customization, templates, etc. I used this for a while, it was a simple of keeping my collection in order and then exporting and uploading to server. Some include automated uploading so you don't have to use FTP yourself.
During that time I created custom export files (you can do so from your software) and from that I build a website that used that exported files with filenames, descriptions, tags, etc as the main database for the gallery. I also write code so that was easy, if you can code that's piece of cake, if you don't, this is quite easy for someone else to do it for you.
I don't remember all of the software I played with over that period of time, some were free, some were not. The good thing is the exporting allows you to get any hosting you want and just upload, nothing ties you to any kind of specialized service (free or not like Gallery3), I did think about that when I was building my gallery, it was a concern over the years and yes my gallery was meant to work for years, I started that around 1998, imagine that.
Anyway, while I don't remember all the names of those options and I have no clue if they are still active, I do know one that I enjoyed a lot: Cumulus, it's from Canto, they used to have a free version, now I know it's paid with cloud option. They are a serious company, check it out.
If you are wondering what my final option was: I developed my own code. There are some very old threads around here where other members mention they did the same thing, building their own option. Sure it can be a difficult path depending on your situation, or you can also pay for it (and hire someone) but the risks of not doing so are exactly what you mention here. I've seen free online options that ended up dying over time.
Other than that you could stay with the version of Gallery that you enjoy the most and keep it kind of frozen on your own server. Good luck.