Hopefully, someone with more DeathRun experience will be able to make more sense of that.
You have many miscellaneous errors that for each one, could mess up other stuff.
Have you updated your server to latest Gmod using the method your host provides, or if they provide manual, using SteamCMD?
You've got
1) several gma workshop addons that aren't loading.
2) several NULL text errors from some DLL_Messageend
Saw a few other miscellaneous too, some occur when spawning.
Seems you have a huge mess of errors of which any could be causing what you are seeing.
Start with no addons or workshop subscriptions.
None.
Only DeathRun, and see if it still works.
If it doesn't, ask/find where support for the DeathRun gamemode is. Surely someone has come across your challenge.
Make sure there hasn't been a deathrun update since the Gmod updates over the past month.
If it does, start adding back addons 2-3 at a time.
Try again, when it breaks, you know it was in the last 2-3