[BUG] Getting the horse to follow you across maps/gamemodes

Horses can remain enabled after switching away from a map/gamemode where horses are allowed.

Expected behavior:
- New gamemode settings should not keep horses enabled by default.
- Horses should only appear in gamemodes where they are explicitly allowed.

--- Original GitHub issue (auto-fetched) ---
Title: [BUG] Getting the horse to follow you across maps/gamemodes

Body:
**Describe the bug**
If you go to Outside the Walls, then open the pause menu with escape and switch maps, the horse will follow you to other gamemodes once you've restarted the round.

**To Reproduce**
1. Go to Outside the Walls
2. Spawn in as a Human
3. Press escape
4. Click on Game Settings
5. Change the map to something else (not Trost)
6. Click change
7. Exit out of the menu and type /restart (or T if on singleplayer)

**Expected behavior**
The horse should only appear on gamemodes where it is allowed.

**Screenshots**
[Image attachment — alt='Screenshot (526)']
Description: (image could not be described: OpenAIError: Missing credentials. Please pass an `api_key`, `workload_identity`, `admin_api_key`, or set the `OPENAI_API_KEY` or `OPENAI_ADMIN_KEY` environment variable.)
[/Image]

**Log file**
https://www.dropbox.com/s/s6px6hgzou3786p/Horse%20across%20maps.txt?dl=0

**Contributor instructions**
1. Try to recreate the bug via the reported reproduction steps. If the bug can be reproduced, remove the investigation label. If the bug cannot be reproduced, please add a comment, attach "On Hold" label and inform the submitter that the bug cannot be reproduced. If there is no reply after 2 weeks, the issue can be closed.
2. Follow the default development team procedures (Create a new branch, apply your fix, create a pull request, submit new builds for testing team in dev squad)
3. Once testing team has confirmed the issue is resolved and the pull request is accepted, the issue can be closed.
