Doesn't do enough extra to justify the extra cost, IMO.
In matchups where you need Cage, you need it in play ASAP, ideally while simultaneously layering additional hate cards or answering additional threats like Hollow One that Cage doesn't answer. Paying an extra mana for Cage is not compatible with these plans.
We’re on a long tangent away from the Lotus, but it’s an interesting question how widely Vintage decks would replace duals with “painless City of Brass,” if that were printed.
My guess is that in many decks the fetch lands and duals have value beyond mana fixing (source of shuffling; wasteland resistance; fuel for Gush and Mystic Sanctuary, etc) so that the duals would still be heavily used.
I highly doubt people will actually pay 17-19 lives for this on turn 1. You'd need a god draw with 3-4 mana floating after the wheel to have a somewhat reliable kill. If you don't win that turn, you're pretty much dead to anything.
You can also cast a mini-Tendrils to gain the life needed to cast the second Wheel. Presumably you are running the Wheel in a TPS shell (since if your deck is based around artifact acceleration only, PO is just better than even a one-sided Wheel of Fortune, no?).
They are definitely replacement effects:
614.1d Continuous effects that read “[This permanent] enters the battlefield . . .” or “[Objects] enter the battlefield . . .” are replacement effects.
The real question is whether they are self-replacement effects:
614.15. Some replacement effects are not continuous effects. Rather, they are an effect of a resolving spell or ability that replace part or all of that spell or ability’s own effect(s). Such effects are called self-replacement effects. The text creating a self-replacement effect is usually part of the ability whose effect is being replaced, but the text can be a separate ability, particularly when preceded by an ability word. When applying replacement effects to an event, self-replacement effects are applied before other replacement effects.
From a strict reading of the comp. rules, the Guildgate replacement effect that causes it to enter the battlefield tapped cannot be a self-replacement effect, since lands are not spells and don't enter the battlefield as a result of "a resolving spell or ability." There is a slightly stronger case for the "enters the battlefield tapped" abilities of permanent spells being self-replacement effects (replacing the permanent's own effect of entering the battlefield on resolution) but since 614.1d seems to clearly label these abilities "continuous effects" and self-replacement effects are "not continuous effects," I think this case is also flimsy.
So: we will get an official ruling before the set releases (including, potentially, clarification in the comp. rules themselves) but right now I'd bet on the controller of the permanent choosing whether it enters tapped or untapped.
This card has no Vintage implications, but is a notable printing nonetheless as the closest replica yet to MtG's most iconic card:
Jeweled Lotus 0
T, Sacrifice Jeweled Lotus: Add three mana of any one color. Spend this mana only to cast your commander.
I don't play a lot of Commander, but naively I expect this card to dramatically increase the variance and power level of that format...