So, I don't think this card is really vintage playable, although it is really cute and perhaps has angles to shoot. (Kismet on legs! Stasis is back BABY).

No, the reason I posted this is because this forum has some of the most intelligent folks in Magic, and I'm very unclear on how this works from a rules perspective. The question for me is what happens when you have Archelos out and play something that ETB tapped like, say, a Guildgate. Both cards have text directing the card to enter the battlefield tapped or untapped. So, what happens?

Assuming they are both replacement effects, I think the answer is that the card MUST enter the battlefield untapped. Rule 616.1 says that if two replacement effects try to apply, normally the player controlling the permanent being effected chooses. But, if one of the effects is on the effected card itself, then the self-replacement effect must be applied first. Rule 614.15. So, the Guildgate's own ability tells it to enter tapped, and then the Turtle's tells it to enter untapped, and so the latter one controls.

But, it's not clear to me that these are replacement effects at all. They're not written in the form "If a permanent would X instead it Ys." Has anyone encountered this before?

Note that some cards like Thalia don't raise this issue because cards traditionally do not say "This card ETB untapped" in their card text, so there is no conflict between effects to resolve.

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.

You control the card which the replacement effects affect, so you choose.

EDIT: Evouga cited the more complicated path to that conclusion. It's quite clear in 616.1:

616.1. If two or more replacement and/or prevention effects are attempting to modify the way an event affects an object or player, the affected object’s controller (or its owner if it has no controller) or the affected player chooses one to apply, following the steps listed below. If two or more players have to make these choices at the same time, choices are made in APNAP order (see rule 101.4).