Should dbatools be installed on all SQL Servers
In general you should be able to use most of the dbatools functionality from your laptop (or whatever host you use to connect to your server via SSMS). If you're the kind of admin who prefers to RDP onto your servers and use SSMS locally, then maybe try to break that bad habit; but in general you should be able to use dbatools alongside the rest of your administrative toolset.

If you've never used powershell at all in your environment before, then you may need to coordinate with your sysadmins or networking team to debug some PS Remoting  growing pains or open up ports here and there. As well, if you want to use dbatools functions as a part of SQL Server Agent Jobs, you will need to install (and upgrade!) dbatools locally (or in a way that the module can be loaded by the agent). I recommend this guidance

And as always - for any specific issues, the SQL Community dbatools slack channel is the bee's knees.