Could you please share the screenshot of user's permissions tab, it will help me to debug.

I feel if you add those users to OOTB content-authors user group as shown below then they will be able to use pick button as expected -     I have tested at my end. Please give a try. 

Is there specific reason you want to use both Classic and Touch UI while authoring? Just wanted to understand authoring challenges here.

Sharing few examples -   https://www.flexibledesigns.rs/creating-a-link-rewriter/ https://wttech.blog/blog/2019/how-to-use-sling-transformers-in-aem/   Here you could see, they have rewriter under /apps/APPNAME/config

What is the functional significance of private folder [path: /content/dam/private/time-management.jpeg]. I have seen in your logger statement.

While uploading asset via AEM Desktop app I am seeing INFO below - 05.02.2023 17:59:53.436 *INFO* [127.0.0.1 [1675600193357] HEAD /api/assets/we-retail/en/people/mens/aem-gems-012016-oak-lucene-indexes-async-local.pdf.json HTTP/1.1] org.apache.sling.engine.impl.SlingRequestProcessorImpl service: Res...

Could you please open this page using Page Information --> View as Published then you may see some exception on page and check the log file at same time error.log.    Please post the exception here.

Could you please refer this, hope this will give you some idea -   https://adapt.to/content/dam/adaptto/production/presentations/2020/adaptTo2020-AEM-Virus-Scan-Roland-Gruber.pdf/_jcr_content/renditions/original.media_file.download_attachment.file/adaptTo2020-AEM-Virus-Scan-Roland-Gruber.pdf

Yes, I used ACS commons with customized page property also and it worked.   In result column you need to mention this (giving example) - Heading Field Label [Some meaningful name] of custom property Property jcr:content/cutomproperty   Hope this helps.

Sharing my personal experience. In my last project one of our admin team members used to make reindex property true directly on environment.