I don't think they should be either; the artificer should definitely be its own class.
Really! Can you go into a little more depth on this? They give out a "wizard subclass" vibe to me ("I contribute to a party primarily by using arcane magic via items and equipment"), though they feel like a better fit proficiency/HP wise with bards (or, as you did, clerics). I feel like they'd need a big unique hook to draw me in as a unique class, and that's something that the artificer hasn't had in previous iterations. 3e's Artificer identity was primarily "I make items and cast arcane spells," 4e's Artificer identity was primarily "I heal people and cast arcane spells," and I would need some distinct story reason for them to not just be a wizard whose specialized knowledge was applied to stuff rather than books and words and wiggling your fingers about. Like, what would define their subclasses, narratively speaking?
I feel like the experience of playing an artificer in 5e -- with its looser, more chaotic magic items -- has a lot of potential to be distinct, regardless of its mechanical chassis. Lots of potential there to ditch the overly detailed accounting of 3e and embrace "The DM Gets To Screw With You" as a condition for making elaborate magic items. But artificers should probably have more reliable construction rules for their "basic items" (ie, class features, spell slots, whatnot) than currently exist.
Hmm...that could get a big mechanical distinction down. Rather than "casting spells," artificers do something rare to find in classes in 5e: they lay down lasting buffs. You have an artificer in your party, everyone gets their weapons on fire all day long, or gets to regenerate for the whole encounter, or whatever. In fiction, this is the artificer enhancing your equipment (making your weapons flaming and turning your armor into regenerating gear) -- doesn't last forever, but it gets the job done. Maybe they can use various "potions" as their daily spell equivalents (throwing alchemists' fire = fire bolt!; healing potion = cure light wounds!). Again, doesn't last forever, but it's
sharable, which means the artificer can give the party rogue a potion of darkvision for when they go scouting, and can give everyone a healing potion before they set out for the day...
...still not sure what I'd see as "subclasses" of artificers, but that might be a bit enough mechanical fob to hang a class off of.