'Remote Getaway' chat message appears before the player teleports through the gateway

The Bug

The 'Remote Getaway' chat message displays when the player touches the end gateway block (i.e. using an Ender Pearl). This is inconsistent with the advancement popup, which displays when the player lands on the other side. This causes the chat message to appear earlier than the popup.

Note that in 1.20.6, the advancement popup and chat message show at the same time: when the player lands on the other side of the gateway.

See this video (unlisted). I've provided a normal and slowed clip for 1.21 Pre-Release 3 and 1.20.6 in the video.

How to Reproduce

Ensure that the gamerule announceAdvancements is set to true.

1. Create a new world. It's better to do this than to use a pre-existing world so that all the chunks are newly generated, making the bug more noticeable. However, you can use an existing world.

2. Run */execute in minecraft:the_end run tp @s 5 70 5* to teleport to the End.

3. Run */kill @e[type=minecraft:ender_dragon]* to kill the dragon.

4. Use an Ender Pearl to go through the gateway that should be generated near the edge of the main end island.

5. ❌ Note how the chat message displays as soon as the Ender Pearl touches the gateway, but the popup displays when the player lands, making it inconsistent.

6. ✔ In 1.20.6, the chat message and popup display when the player lands.

Additional Notes

Related to MC-272814.