Oh wow, the conversation has really taken off...
So in conclusion all you’re really getting here is a cantrip 4/4, which is good enough for limited, but not Vintage.
I think it’s probably good enough in Standard as well.
The card has already put up league and tournament results in Pioneer and Modern. It is unsurprisingly quite good with Uro, Growth Spiral, and fetch lands.
As for Standard, uhhh...
Uro is certainly the most powerful card in the format, but Omnath has undoubtably established itself as beyond "probably good enough".
One particular interaction that @Botvinik and others seems to be missing is the ability can be triggered on the opponent's turn so you don't have to make mana off a fetch if you aren't using the mana. The key synergy that hasn't been explored in Vintage or Legacy is with Sylvan Library where you turn the 7 life from a fetch into 2-3 cards a turn. This card has the making of a control finisher. Along those lines, it has a lot of value pitching to Forces in those control decks.
Edit: If people want comparisons, is this card that different from a massively power crept Bloodbraid Elf? It draws a card when it comes into play and gives you the mana to cast it if it's a spell. There are differences of course, but the value and mana generation are similar. You also have additional utility in the life gain and the damage.
Ummm lots of different stuff can be cast, like Treasure Cruise, Dack Fayden, Wrenn and Six, Oko, Lightning Bolt, ect. Casting 1 spell a turn is not really how Vintage turns are played.
Then I want to ask a specific question what are you planning to spend the mana on? Because you have already raised your curve by running omnaths so what 5 drop is worth playing that you think fits into the deck and why does your 4 drop need to be chased by a 5 to be effective. Now if the mana stuck around so you could spend it on say a force of will during your opponents turn I would agree but it empties witch means you need to be playing a 5 mana sorcery speed threat to follow omnath to keep his mana from just exiting the pool. And if Omanth refunds just to have you not spend the mana, he may well have never given you a refund in the first place.
This is why I could not make this deck work when I tried. He is not enough on his own and whenever I cast a 4-5 mana walker to follow him up I wonder why I did not skip the middle man and run an individually game winning threat like oko instead of this weird 4/4. Given cantripping helps no question, but when I cast this my next thought was usually I wish this were either jace or oko.
Karn was absurdly broken to the point of restriction. This card is obviously no where close to that, nor are any of these other 4 cost cards or planeswalkers for that matter. Even Narset that was also restricted pales in comparison to Karn's power level. One of those cards that they definitely pushed super hard.
I don't think Jace is bad, but it seems outclassed by other planeswalker options now. Decks are mostly just playing it because its less colors as you've indicated, and they want that versatility that a planeswalker brings.
As the discussion is on Omnath, like I've said, I think his/her playability would stem from the fact that its not actually costing you any mana at all.