TL;DR
My system storage size would not stop increasing on my Android 12 Pixel 4a (similar situation to the OP). I was able to reclaim it by saving large dummy files into the free storage (on the user side, internal storage) and then removing them again.
Detailed instructions
Pick a large file you have at hand: movie, Linux ISO, Android ROM, etc. Create multiple folders inside the internal storage with dummy names and copy the same file into each one of the folders. You will see the system storage size shrink as you use more and more space on the user side. Once the system storage has shrunk a bit you can copy the next batch of files. Rinse and repeat.
My experience
In my case I started with a system storage using 72 GB of my 128 GB memory and ended up with only 14 GB of system storage, hence freeing almost 60 GB. I found out that Android would free 1 GB when the storage is full so I worked with around 700 MB of files at a time.
How to optimize the process
I found out that using a FTP server on my phone and uploading the file from my PC slowed down the process in a way that Android could keep up with the copy and free enough storage in time for the next file. Hence I prepared a large amount of files on my PC and let it run for one hour, until a warning popped up that my phone is running out of storage. I had to fiddle a bit because many apps would crash at this point but after freeing some storage the phone resumed normal operation.
My theory on why it works (at least for me)
The rationale behind this mechanism is that Android (at least recent versions) will make use of all the free storage it can grab for its internal use, especially cache for system components. Many people already hint at this behavior on the Internet and recommend clearing the app cache, which I did with no result. The catch is that while the user can empty the cache of user installed apps the cache of system components cannot be cleared without rooting the phone or doing a factory reset and none are practical for daily use. I guessed that Android will give up that storage whenever the user needs it and it worked.
Last words
So at the end the system storage size continuously increasing is probably not an issue which requires fixing. It's unfortunate and inconvenient though that Android shows the internal storage as full when it's actually not and presses the user to delete "useless" pictures and videos instead of cleaning up all the mess inside the system storage.
References
This issue is documented in different places: