Title. Suppose the mobile device is well supported by both.

I suppose if it's not a Pixel device, I'd say LineageOS.

eOS to lineageOS is like Ubuntu to Debian; a more newb-oriented solution with some stuff preinstalled but with less choices; e.g. you can't switch out the iOS-like launcher. if you know your way around, you're wasting your time with eOS. but if it's for a less technically adept user, you might be doing them a favor with eOS. the murena suite can be opted out of.

You can install a custom launcher on eos. I will say that the default apps cannot be uninstalled, but I heard there's a way to do it by connecting your phone to your PC.

The e/os Advanced Privacy feature is very nice, great way to see which apps have tracker whilst blocking them. Same as running DNS filters but built in experience, easy to view

In itself LOS is fantastic but needs be combined with something like reThinkDNS or another DNS filtering option

Rethink on LOS is fantastic, in particular if you can do the whitelist method - block everything by default, and permit only that which you approve. Takes some time & effort but great control is the reward

I prefer layers, example LOS w/rethink DNS all going thru Control D or NextDNS with Hagezi lists enabled

Overkill? Perhaps , but I love the reporting , helps in learning what's going on, and apps/sites to be avoided

Opinions don't matter. Just check how updated they are. This impacts security and thus privacy.

I've come across an article recently that mentions both (and others), it might interest you:

https://blog.gridranger.dev/mobile-oses-featuring-fairphone-5/

I can't really compare them because I've not used LineageOS, but I got a Fairphone Gen6 with /e/OS a few weeks ago and it's better than I expected.

One of the problems with custom roms is you have to unlock the bootloader to install them which can then prevent some apps from working. I didn't have a problem with that because the phone came with /e/OS pre-installed with the bootloader locked. Depending on what you need the phone to do it might be worth considering. It gets quite frequent updates.

The main issue I had with /e/OS was the launcher was too basic for my needs. However, this was easily fixed by just installing a different one. Most of the default apps are good enough that I haven't needed to find a replacement for them.