With religion I think we largely just go with "this is what the group identifies themselves with / as" and leave it as that.
Arguably the politically active "religious right" in the US has long since abandoned most everything Jesus had to offer. We still identify them as such.
The problem with this approach is that there are also parts of the mainstream Christian doctrine (both modern and historical) that don't have Biblical basis, Trinity being one prominent example. One could go further still and reasonably argue that most of mainstream Christianity is basically teachings of Paul rather than Jesus.
And yes, there are also valid arguments against either one of those points. But to meaningfully engage in such a discussion, one needs to be well-versed in theology - more so than the vast majority of people who self-identify and are normally identified as Christians.
Eh, I don't think the "it's all based on Paul" can be a reasonable argument at all. Even when people do argue it, they only ever allege the same 2 points, which even assuming those points are valid would only be a tiny fraction of Christian belief, both theoretically and practically; you don't need theology for this. Of course, to those whose sole exposure to Χianity is Internet arguments this might not be obvious.
The whole focus on "Trinity" is weird but makes a little more sense if you realize it was originally developed to stand against certain heresies, but later mutated (often re-entering those same heresies again). This one is indeed theological ... or perhaps we need a word that is to theology as historiography is to history.