Yup. That’s a noble goal, but the underlying technology doesn’t have a “test connectivity” function. So it’s going to have to be just out-and-out attempting to access something, and trapping the error. What’s more, you’re going to actually have to try and read or write a file, because simply mapping to the share won’t confirm that file operations will work. There’s just no “ping” function like this in the SMB APIs. So mapping a drive is a good first go - but it won’t necessarily confirm that you can do anything.
But that won’t confirm that Get-WmiObject will work, or Get-WinEvent, or Get-Process (which uses the Remote Registry Service). Each of those APIs might deny you, even if one of the others worked.
Now, look, if your assumption is, “I’m actually just wanting to test if I have local Administrator privileges, because if I do, I can do all of those things.” That’s something you can test directly. Exactly how depends a bit on how you think you’ll become a member of that local group on each machine, but it’s probably easier to test that one assumption that to individually test every single possible API that allows you to connect.
Of course, the alternative is to ditch all that crap and just use Remoting. One check (and there’s a Test- cmdlet for it!) and you’re in or not, period.