The Artificer class released as Unearthed Arcana has the same spellcasting progression (meaning number of spell slots at certain levels) as Eldritch Knights (PHB pg. 75) and Arcane Tricksters (PHB pg. 98), even with regards to gaining the spellcasting ability at 3rd level as opposed to 1st or 2nd level like all other spellcasters. I have a player (I'm the DM) who wants to make one and we'll be starting at level 3 (they'll be going the Gunsmith route, in case that's relevant).
However, whereas the two aforementioned archetypes get access to the whole Wizard spell list (albeit with spell school restrictions for most of their spell choices), Artificers instead gain what appears to be a completely arbitrary list of spells, mostly based on the Wizard's list as far as I can tell, but with many omissions and a few additions, such as cure wounds, which is not usually available to arcane casters (outside of Bards and certain archetypes such as Celestial Warlocks and Divine Soul Sorcerers).
I have no idea why these spells were chosen, but my question is not "why did they choose these spells?"; they chose them for whatever reasons, and I don't like their choices.
So I was going to instead ignore the Artificer spell list completely and simply allow my player who will be playing an Artificer to pick from the entire Wizard spell list, similar to Eldritch Knights and Arcane Tricksters; this makes sense to me since Intelligence is their spellcasting stat. I was also going to allow them to have a couple of cantrips (where previously they couldn't) in line with Eldritch Knights (two at first, an extra one at 10th level).
My question is: what impact will this have on the balance of this class?
For example, this class would have able to cast cure wounds, and now will not be able to (so could not serve as the "party healer", for example; note that there is no party of which to speak of yet, so don't ask "what will the rest of the party be" because I don't know that yet). And of course, access to "at will" cantrips and, at higher levels, spells like fireball. I don't think there are going to be any major impacts (I believe this class is balanced well enough, unlike the Mystic), but are there any impacts I'm overlooking?
Note: I have no prior knowledge of Artificers in previous editions of D&D, nor any knowledge of the Eberron world (since this came up in comments and is related to Slagmoth's answer). Hence my impression of the class from a flavour/lore perspective is entirely as gleamed from that UA article.