Revert "Switch to updating presence via /sync calls instead of PUT /presence"

[andybalaam]

This reverts commit 7e57394. Original PR: #11223

Reverting because we need to wait for a fix for matrix-org/synapse#16039 first.

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This change is marked as an internal change (Task), so will not be included in the changelog.

[RiotRobot]

The backport to staging failed:

The process '/usr/bin/git' failed with exit code 128

To backport manually, run these commands in your terminal:

# Fetch latest updates from GitHub
git fetch
# Create a new working tree
git worktree add .worktrees/backport-staging staging
# Navigate to the new working tree
cd .worktrees/backport-staging
# Create a new branch
git switch --create backport-11365-to-staging
# Cherry-pick the merged commit of this pull request and resolve the conflicts
git cherry-pick -x --mainline 1 bbab65a4cdcc51b65a6def388cf46b682e4914cc
# Push it to GitHub
git push --set-upstream origin backport-11365-to-staging
# Go back to the original working tree
cd ../..
# Delete the working tree
git worktree remove .worktrees/backport-staging

Then, create a pull request where the base branch is staging and the compare/head branch is backport-11365-to-staging.