Teleroam
The Mac stays where it is. You reach it from your iPhone, or from a browser on a computer that isn't yours.
The Mac is in the other room. AirPlay from that Mac to the TV in front of you, then pause, skip, and pick something else from the phone. The show is playing from the Mac, so the phone is still yours. AirPlay from the phone itself takes the phone over for as long as the video is on the TV. You can't text, or look something up, until you stop it.
Add a keyboard and a mouse. The screen in front of you is the Mac's desktop: your apps, your files, the accounts already signed in. The phone is the connection. The computer you're using is the one at home.
Your Mac already has your iMessages and texts. On a work computer, open them in the browser and answer from there. They stay on your Mac.
The document in your home folder, the photo you didn't sync, the project you were halfway through. Open it right where it is.
Email, notes, banking, your password manager. They're already signed in on your Mac, so you never have to log in on someone else's computer.
Your Mac is on your home Wi-Fi, so you can reach the printer, the file server, the router page, or the cameras that only work at home.
A work PC, a library computer, a hotel lobby. Open a browser page. Nothing gets installed, and nothing stays behind when you close it.
Personal things stay on your Mac instead of spreading across a work laptop. Close the tab and it's gone from that machine.
A download, an export, a backup, a render. See if it finished, or stop it, without walking back to that room.
A license tied to that machine, a local tool, a window you left open. There is no phone version. Open it on the Mac, from wherever you are.
A permission dialog, an update, a prompt that says it needs you. Answer it from the phone instead of going to the other room.