Only if you use the same account on both computers. What means the GP will have to use his work account on his personal computer just to jump into another machine.
Because their colleague wants to let them drive?
Because my sister wants some help installing a program?
There are so many situations where using a computer doesn't involve strict security protocols and heirarchies. Sometimes MS gets a little too caught up in their corporate environments.
And then all your hypotheticals speak to issues which don't involve RDP, at all, which I think shows where this disconnect is. RDP is not about sharing a currently active desktop session between two or more users, so all the hypotheticals you shared aren't the use case for RDP. RDP will transfer that console session to the new RDP connection. So say Alice is signed in to her computer locally, and then wants to ask Bob for some help. She shares the hostname for her computer, and tells Bob their login (a terrible idea regardless of it being a local or Microsoft account). Bob fires up RDP, connects using Alice's account, and now Alice's machine gets disconnected from that desktop session and is sitting at the login screen while Bob now has Alice's desktop session.
This doesn't matter if its a Microsoft account or a local account, this is just how the RDP protocol works on Windows. If you're wanting to have a screen share with the built-in Windows tools, the tool for that was Remote Assistance or Quick Assistant. The usefulness of that tool doesn't change whether using local accounts, domain accounts, or Microsoft accounts. Or just use a different tool altogether, of which there are many.
So, the OP is either having problems using a cloud user RDPing into a domain, or connecting his computer into the work's domain and using the same user all the way.
Here is where it gets complicated, what you say is true for personal accounts and Hybrid Azure AD accounts, but it becomes super difficult with pure Azure AD accounts.
Also with pure Azure AD accounts you have to add each users permissions manually to each machine they need to login to since Azure AD groups don't sync down to each machine.(Correct me if I am wrong here, it could have changed since I last did a pure Azure AD install 3-4 years ago)
I haven't tried but I think these days you can actually authenticate RDP with a security key associated with the account.
I'm of the exact opposite persuasion. RDP'ing into a computer with your Microsoft account is easier, along with file and folder permissions across network shares and other account related things.
I used to have to manage an AD domain to have the same set of accounts across all my various Windows machines. Now that I can sign in with my Windows account, its the same account across all my machines. Permissions are easier on network shares, its the same account. Friends that come over and want to use my machines can log in with their own account. Sharing files with them is then just granting access to their Microsoft account. Its pretty much entirely replaced the desire to run AD at home, which IMO makes things loads easier.
As for your password being out of sync, I've only experienced an out of sync password in cases where the device could not have an internet connection. Once the device was able to get an internet connection, it prompted for me to refresh the current device credentials (lock and log back in), and it then authenticated against the cloud Microsoft account. In the 10 years and ~30 different machines used I've yet to experience a single real frustration of the online account and local account getting out of sync for more than a single password change when literally in the wilderness. In which case, it was just the last password I used to log in to the machine, and then updated when it got network connectivity again.