Riding a mob into powder snow no longer affects the player properly

This behavior was introduced in 1.21.6-pre1.

The bug

When riding a mob into powder snow, the freezing overlay appears on the screen as expected, but it does not disappear after leaving the powder snow. The overlay stays on the screen indefinitely, even though the player is no longer in contact with the snow. It only goes away after reloading the world, re-logging, or walking into powder snow on foot and exiting it.

Steps to reproduce

1. Join a Superflat world with commands enabled.

2. Run the following commands to fill a nearby area with powder snow, summon a saddled pig, and give yourself a carrot on a stick.

   /fill ~ ~ ~ ~31 ~31 ~31 minecraft:powder_snow
   /summon minecraft:pig ~ ~ ~ {equipment:{saddle:{id:"minecraft:saddle"}}}
   /give @s minecraft:carrot_on_a_stick

3. Mount the pig and take the carrot on a stick in your hand.

4. Ride into the powder snow without pressing the forward key.
   → ❌ The pig will start shivering and take damage as expected. The player, however, will not see the powder snow overlay and will not take damage.

5. Press the forward key.
   → ❌ The powder snow overlay will now begin to show up for the player. However, the player will still not take damage, and if you stop pressing the forward key, the strength of the overlay will not increase anymore.

6. Ride out of the powder snow and dismount the pig.
   → ❌ The powder snow overlay will remain.