While everything /e/OS users can say about this will be helpful, there is an objective situation which you need to sort out for yourself anyhow when considering /e/OS or, in more general terms, limiting Google dependencies on an Android phone:
Most Apps which exist now have a certain level of dependency on Google services (background processes on a phone with the usual preinstalled Google stuff) built in.
The key question is:
How dependent are you on such Apps working 100% 24/7?
/e/OS includes an Open Source alternative (called microG), which overall does a very good job in mimicking some of the most needed Google services on the phone to enable Google-dependent Apps to work despite the genuine Google stuff missing. Most of those Apps should work fine.
But not all of those Apps work, and for such Apps which work at a certain point in time, this could break at any time, conceptually as well as observed in reality.
From what I’ve seen so far I guess it is more common for problematic Apps to show problems from the start rather than to first run smoothly and only run into trouble later, but the latter can really happen (seen it myself), and if you are affected it doesn’t help you how improbable this might have been. (I knew this could happen, so it wasn’t a big deal for me.)
If you rely on Google-dependent Apps working 100% for your job, for medical stuff, for any use case you really, really need any such App to work for … don’t try your luck and stick with the stock vendor OS (there can be enough problems with this also as it is, regardless of OS and vendor, but at least there’s a professional business chain of responsibility there to try to fix things).
This leaves the question how to find out potentially problematic Apps.
Aurora Store directly (and anonymously, if chosen so) accesses the Play Store and shows whether Apps are GSF-dependent (GSF = Google Services Framework) in its search results.
This doesn’t say how deeply Google-dependent the App in question is, it may not mean anything in practice or it may mean the world for this App, but it is a prime hint to be careful with this App, if in doubt.
Else there are lists such as [LIST] Apps that work or do not work with microG - Lists - /e/OS community and [LIST] Banking Apps on /e/OS - Lists - /e/OS community … but keep in mind that a Google-dependent App being observed working at a certain point in time may not be working at another point in time.
Why not stay with the stock vendor OS?
In case alternative Android OSes turn out not feasible enough for your use cases, staying with the stock vendor OS is not the end of the world, there are things which can be done to reduce Google dependencies, starting with not using a Google account on the phone (which is possible, you can skip the request for this in the initial setup, or remove the account later in the Settings) and not using the popular Google Apps.
Here’s an old post about this (might need revising now, but it should still serve as an example of how to possibly approach this) … Google Apps keep reinstalling/ reactivating - #34 by AnotherElk