Try exiting it and then make sure no tmux process is still running, by for example running ps -aux | grep tmux.
For future reference: the command to kill the tmux daemon (and as a side-effect, all other running tmux processes connected to it) is tmux kill-server (or in tmux, typing <prefix> :kill-server, assuming default keybindings).
You don’t actually have to set all the modification dates to now, you can pick any other timestamp you want. So to preserve the order of the files, you could just have the script sort the list of files by date, then update the modification date of the oldest file to some fixed time ago, the second-oldest to a bit later, and so on.
You could even exclude recently-edited files because the real modification dates are probably more relevant for those. For example, if you only process files older than 3 months, and update those starting from "6 months old"1, that just leaves remembering to run that script at least once a year or so. Just pick a date and put a recurring reminder in your calendar.
1: I picked 6 months there to leave some slack, in case you procrastinate your next run or it’s otherwise delayed because you’re out sick or on vacation or something.
My Linux laptop is set to check for updates daily, which I then apply manually when I notice the tray icon. I sometimes procrastinate when it comes to reboots though.
My Android phone is on auto-update, which seems to mean whenever it’s being charging for a few hours (so typically when charging overnight). Because the battery is still pretty good and I don’t need to charge daily, that comes down to once every 2-3 nights or so.
My personal Linux servers (which run my self-hosted apps) are configured to automatically apply all updates (and reboot if necessary afterwards) at the time of day I’m most likely to be awake and available to manually fix stuff if anything goes wrong. The Docker-containers that run on them mostly get auto-updated to the latest version every 6 hours by Watchtower. A few containers have more cautious policies though, ranging from pinning a major version (but auto-upgrading to new minor versions within that) to pinning a specific version and at most sending a notification if there’s an update. The latter is limited to stuff that has broken before and/or where newer releases are known to be buggy or incompatible.
When it comes to major updates (i.e. new distro releases) of my Linux machines, I typically wait about a month before upgrading because I’ve been bitten by release-day bugs before.