I don't think removing the +1 card will do anything other than increase variance.
The +1 Card makes a monster into a mega-monster, I'm positive. Say you play two Untitleds, which for starters is 20% easier to do with the +1 Card on it. You'll draw a total of 3 extra cards by playing those two, which is pretty good. But without it, now you have to have drawn both into your initial hand to play two, and after you do so, you only get a total of 1 extra card.
That's 6 vs. 4 cards remaining in your hand afterward, from which, if you have any action cards at all, be they more Untitleds or otherwise, you're probably going to kick off a megaturn.
The likelihood with which Untitleds can kick off an action chain -- of whatever kind -- is the only real constraint on this thing, and although the weaker version of the card is still stupendous, that constraint is a big one. It's why Throne Room can be affordable: the chance of drawing it dead is a real risk. Similarly, this is exactly why Golem is orders of magnitude better than Village. Both let you play two more actions. Golem guarantees it.
Speaking more abstractly, +1 Card is an ENORMOUS buff on a non-terminal. It turns a simple cantrip into a Laboratory, after all. With +1 Card, Scout would suddenly become awesome.
I think building engines with multiple copies of this seems powerful, but in practice, probably not as much as people assume. 2 untitled = 1 lab. 3 untitled = 3 labs. You need to chain together 4 or more to exceed laboratory, at which point you're probably drawing near your entire deck anyway and it ceases to be much of an advantage.
That's not where the strength comes from. The strength comes from the fact that
any cantrip played after just ONE Untitled becomes a Laboratory all by itself. One Untitled, and those Pearl Divers you picked up with extra buys just because they did no harm are now suddenly a Laboratory chain of their own. Those Villages you picked up? They become Laboratories that turn your
terminals into Laboratories too!
All that, with ONE Untitled. Chain together two Untitleds, which you'll probably be able to do since one can explode your drawing power so much, and I'm pretty sure that's a guaranteed deck-draw no matter how many greens and Curses you've got. Because then your Moats are drawing like Council Rooms, and your Courtyards are doing even better. Maybe if you're REALLY clogged, you need three, but no more. Price this card like Minion, and it won't matter if the split goes 7/3 against you, because the guy who bought 7 is down 4 Duchies that might as well have auto-Islanded themselves for all the harm it did your deck.
I might well be exaggerating here, but I really don't think by much. If you look at it another way and credit the extra card draws to the Untitled directly instead of talking about it in terms of turning Pearl Divers into Laboratories, consider that adding a single Untitled to any kind of engine that's not quite up and running but slowly gearing up (say, Village/Wharf or Worker's/Conspirator or Festival/Library or whatever). Well, that one Untitled might very well be responsible for drawing 4-6 extra cards. That's already enough extra oomph to turn that revving-up engine into a reliably-running one.
But you really don't need to wait that long. Open with two action cards, get two more on the second shuffle, and then you can probably buy an Untitled on turns 5 or 6 and reliably draw it with enough of those otherwise clashing actions to roll.