There is the third way: install mods manually :-)
In any case, one big problem is that ContentPatcher mod is not installed. A lot of mods want it. The other problem (maybe because of using Vortex that can be bad at it but I am unsure): duplicated mods. E.g:
| 09:33:27 | ERROR | SMAPI | - More Mushrooms 1.0.0 because it requires mods which aren't installed (Esca.FarmTypeManager). |
| 09:33:27 | ERROR | SMAPI | - More Mushrooms 1.1.0 because it requires mods which aren't installed (Json Assets: https://www.nexusmods.com/stardewvalley/mods/1720). |
As description of the mods says "Delete old files before installing a new version because file structure changed", I guess Vortex could ruin it on updating. Possible solution: delete both version of the mod manually and then reinstall with Vortex the latest one.
As of other errors: there are a few complains about missing dependencies that you can install to fix. All missing mod names you can get from the log.
After you reinstall Mushroom mod and install the missing mods, try to run the game again to check if everything is fine.
