iOS has a good document scanner built into the files app that creates PDFs. You need to long press and select scan document.
[0] https://wittman.org/blog/scan-paper-document-to-pdf-with-iph...
The scanning part is only the beginning, the phone -> network pipeline needs to be automatic and reliable, and Apple will do nothing for third party synching.
Perhaps a Shortcut pipeline could do it ? If it still gave enough flexibility about batch scanning etc. ?
https://www.techrepublic.com/article/how-to-use-the-files-ap...
I use iCloud to do exactly what you say and trash the physical copies, but it should be possible with anything in the files app AFAIK.
You still need to manually save each document where you want it, instead of the app doing it automically in batch at every scan. With dedicated scan apps I never need to think about it, just scan and I know it's been done.