Nothing prevents a bard from using a holy symbol or having faith either...Well, let's look at those examples. Bards have no lore ties to any sort of faith, so, the lack of things like holy symbols and well, faith, would differentiate bards from clerics pretty hard.
Celestial Pact Warlock is pretty close to a cleric, true, although, again, most clerics aren't sprouting wings. I'd say that a CPW is pretty strongly tied to its own lore.
Wildfire druid and druids in general should be strongly tied to their own setting lore as well. These are servants, generally, of very specific gods.
And an artificer kinda lacks all the religious trappings just like a bard.
So, no, I don't think a couple of shared spells is enough to make clerics indistinguishable from these examples.