How do you extract in Egg Heist?

The word doing the work there is "vulnerable". Clearing your personal objective does not pull you out of the match. It only makes the boat usable. You still have to get to it, and other players are in the Lost Isles doing the same thing, so the run back is where most attempts die rather than the collecting.

Of the two ways to tick that objective, the egg is the blunt one. Acquire and submit a single Aniimo Egg and you are cleared to leave. Eggshell Coins are the fallback for a run where you never get your hands on an egg at all. Either way, once the requirement is satisfied you extract safely at the boat, so the goal is to get the objective done early and then treat the rest of the run as optional.

None of this applies to the version of Egg Heist played exclusively against AI enemies, which you can run solo or as a two-player team with a friend.

How many Eggshell Coins the coin route needs

60,000 Eggshell Coins clears the requirement. The game does not publish the actual threshold, and it was cut significantly in the February 4 update, so the real floor now sits somewhere below what it used to be and 60,000 is simply a figure known to be enough.

That is separate from what coins are for afterwards. Items and resources you bring out of Egg Heist sell for Eggshell Coins, and there is a bulk sell-all interface for clearing the lot at once.

Carrying the central giant egg

The giant egg in the middle of the map is not a personal extraction. Whoever picks it up transforms into the egg and loses their combat abilities entirely, which means your team has to escort you until evacuation. You are cargo at that point.

What you can still do while transformed is partly unsettled. Jumping and dashing are in; the full movement set is not pinned down. The number of three-player teams contesting a single match is not settled either, so how much pressure that escort is under varies.

Not settled: The coin threshold is the loose end here. Because the requirement is not published and moved in the February 4 update, the only figure you can rely on is 60,000, and anything lower is guesswork until the number is stated.