You are understanding that correctly.
I have no idea if the volunteer background check would show the LTC, nor, if it did, if that would be reported.
But once you become someone 'under their control', even as a volunteer, they get to set the conditions. Their control is limited to what you do, or do not do, as a volunteer.
'Is it a legal matter' if you don't obey their conditions? I don't know. With LTC, if the condition you disobey is to carry, the behavior itself is not illegal, unlike, say, doing drugs on a field trip with the kids.
Once again, we're in CA Bill Season.
There is no value at all complaining or analyzing or reading tea leaves to decide what these bills really mean or actually do; any bill with a chance to pass will be bad for gun owners.
Let us simply oppose them all - write, call, attend meetings with legislators and tell them they're wrong.
The details only count after the Governor signs the bills.
Not a lawyer, just Some Guy On The Interwebs.