The main problem with the Overground name is the confusion with the long-running Londoners' parlance of "the overground" to mean "the big trains" in general (especially if you mostly use the Underground). When I grew up, north of the river where Underground use was mostly concentrated of course, any neighbour or family members who was venturing on local trains south of the river would inevitably make a reference to needing to use "the overground". And it's still "regular-person-speak" to this day. On a recent Tim Dunn programme about the Underground, he interviewed a busker who uses various pitches in Underground stations; said busker went on to make a reference to going elsewhere, and leaving the Underground, and made a natural-sounding reference to heading to Charing Cross for "the overground". (Meaning, main line in general of course.)
Indeed - "North London Line" has been normal for generations (along with, at one stage, in some circles, "the magic railway" when it was a much much less frequent service and its actual appearance - when it did turn up - combined with the strange places it went to, seemed magical). Similarly, the East London Line naturally morphed into that when it switched from LT, who had long shown it on tube maps as "Met Line - East London Section" or similar. I hardly know any Londoner who uses them who doesn't use those names for those two lines as often as saying "the Overground". GOBLIN is perhaps slightly less entrenched, but even then it's not just train fans, it's been called that in local press reports about the line for many many years.
If line names were essential (which I don't think they are), it's really stupid not to at least use existing ones which are well established. It's not the opposition to them which is "political".