Problem:
When the latest version of the Toolset Types plugin is activated, it becomes impossible to activate any other plugin, although deactivating them is still feasible. Additionally, installing new plugins also becomes unfeasible.

Solution:

I checked it and the issue is fixed if we install the latest OTGS Installer Version 3.1.1.

There are some dependencies needed in Toolset that are fixed in this plugin's new version.

I installed it and then updated the Toolset plugins. I tested it and I'm no longer able to replicate the issue.

This version will be added to the download page and change logs as soon as possible: https://toolset.com/account/downloads/

For the time being, in order for you to apply this fix in the live as well, I uploaded the latest version of the installer plugin to your webroot, you can connect to it via FTP and download the zip file or download directly in the private link I sent you above.

Then, proceed to manually installing it and update the Toolset plugins afterwards.