Graphene will strongly discourage you from retaining root over your device, while all LineageOS releases present rooted ADB.
I understand that you can forego the final relock of the bootloader and install Magisk into Graphene. How that fares with future OTAs I do not know.
There are other differences that might impact you. The eBay app does not work on Graphene but is functional on Lineage. Graphene's launcher and keyboard are poor compared to Lineage. The Vanadium browser's dark mode is not as good as browsers on Lineage. Pattern lock is not available on Graphene.
Just tried it out with GrapheneOS on my Pixel 7 and the eBay app works fine.
Generally I think the marketing of GraphenOS is interesting. They are usually positioned as the "Absolute security. No compromises." ROM, but in my personal experience, they are the "It must work. No compromises. Then make the rest as safe as possible"-option, given aspects like using the "real" Google Play Services (if desired), but sandboxed, instead of MicroG or unrestricted Google Play, which pretty much all other ROMs roll with.
I've used the eBay app on various Pixels running GrapheneOS since the Pixel 6 without issue. In the Owner profile, in a secondary profile, in a Private Space in the Owner profile, installed via Play, installed via Aurora.
I'm not saying yours works when it doesn't, but there may be something else going on.
I also don't understand this question. The default launcher is like twm on Linux. It's only goal is to give you some initial GUI so that you could install something you actually like.
Your question doesn't make sense, GrapheneOS said that they prefer spending their time implementing and fixing security issues rather than things that can be easily changed from open source offerings. See https://news.ycombinator.com/item?id=47247778.
I've never rooted my phone for LineageOS (and I have installed it on at least a dozen phones now). The only thing you need to do is unlock the bootloader (which you cannot relock, like with Graphene).
Any idea why any app that works on Lineage would not also work on Graphene, provided you are okay enabling the exploit compatibility toggles where necessary?
Interesting that this stops people. It could be correct of course, I just can't think of anything that would cause it to work on Lineage but not Graphene. They're very fast about upgrading to the latest version, that can't be it, so then... on identical hardware... what could possibly be incompatible all of a sudden? I'd just ignore the warning and try it out!
If you need to ask, probably graphene. IMHO the only downside to graphene is that they consider the user to be an attack vector and therefore eg. actively object to allowing root access, while lineage is less bothered by it. If you don't care about that, use graphene. If you do, probably go with lineage.
Didn't lineage remove built in root years ago? That means you still need a third party root solution like magisk. Factoring that in there's not much practical difference between lineage and graphene.
I believe LOS still does userdebug builds, so you get root over adb by default. It is pretty much impossible to create a usable local backup on official no-root Graphene builds with the recommended re-locked bootloader.
I switched to Graphene with my new phone. It includes SeedVault which is unable to backup most of my apps and ADB backup works with even fewer (deprecated by Google). So only things I could sort of backup were apps that had their own config/data export options. The stupid, "we know best" restrictions from being able to copy between user profiles also made backup and restoring a PITA (upstream android stupidity, not Graphene specific) It was a few days later before I was back to where I was before having to restore stock OS to get a warranty repair (GrapheneOS does not include repair mode or some such thing that the Google service place required). If I was still on a rooted phone, it would have taken a few minutes to restore everything.
With one minor caveat[0] yes I believe neither actively includes it, but GOS devs actively denounce being able to give apps root access and at the very least I've never heard LOS devs say anything similar. While a quick web search will turn up discussions/repos/articles about installing magisk on GOS anyways, they give me the impression that it's more likely to break; in particular, GOS (understandably) wants to re-lock the bootloader and that's not going to play nice with things that (necessarily) modify the OS. So like... I think you can root GOS, but heuristically I would be much more nervous about it breaking.
That said, for a user who says "I want a simple easy to use replacement for google captured android", I don't think rooting is likely a concern, so taking the more polished GOS is probably desirable.
[0] I don't have a non-magisk LOS phone to check, but I think stock LOS includes an option in developer settings to allow root for adb. This is mostly not what people mean when they talk about root access, but it's a form of root access.
Personally I am not sure what "replacement for google captured android" means. That said, if you aren't using a pixel, you aren't using graphene. If you do have a pixel, then graphene is the way to go.
I find it easier to install, and the fact that you can run google play services in a sandbox, is a great safety blanket.
