With older symfony apps there is the concept of environments. Symfony used to come with a seperate controller for 3 environments (prod [production], dev [development] and test [unit tests]). So I would expect that you are configured as production, which means that your app is running the app_prod.php front controller. This will be setup to operate in the same way that apache will often be configured to run by default an index.php file if you access a webspace path directly as in http://www.somesite.test/. With Mautic, things have been configured to run the app_dev.php frontcontroller for every application request.
The other thing you have to understand is that symfony generates lots of code, which is what actually gets run. This involves twig templates, doctrine models, routes and lots more stuff. Symfony comes with a console app that has various command that let you generate this code, however, it may be that the apache user doesn't have effective read perms on the files the OS user generated.
So my simple fix would be to do this:
-have script delete the entire app/cache/* contents recursively, which your bash command list does. Prior to doing this you might want to explore the contents of that dir. You will have a directory for any environments that ran, which I would expect ideally would only be a prod directory. As you can see in the logs you showed, the runtime is trying to access some doctrine orm model proxies that couldn't be opened.
Assuming this is the issue, once you delete the directory, open the Mautic app. This will cause all the code generation to kick in and all these files and directories will be owned by the effective web user. That user does need the ability to read/write/execute in the app/cache directory. It might take a few seconds for all the code generation to occur as you are essentially "warming the cache" manually, but once the files are generated they won't be generated again and everything should run at full speed.
You could also add your own web function that would clear the cache from the web app, but if the web app is in a situation where it doesn't actually own the directory and/or files in question, there is no way for that user to fix anything once it's broken, and only the account that owns the files will be able to delete/chmod/chown them (or via root or sudo).
Assuming you deleted the app/cache/prod user, and ran the Mautuc app, you shouldn't encounter any problems. Not knowing what came with Mautic, it could be that there are scripts being run on some schedule that undo the issue you will be fixing, but removing the app/cache/prod directory should fix the problem.