I'm quite confident something could be invented/improvised to take the place of the actuator motor, but...
By the time you put that much effort into it, and that much time scrunched up underneath the dashboard, you really could much more EASILY replace the broken motor.
Check with you local car salvage / junk yards and I expect you'll find a ready supply of used ones for $5. Even a new one is $30 or so.
See "Related Questions" below for more on how to replace the actuator - it should give you an idea on where it is and jump-start the process of bypassing it, or fixing the broken one.
Deep under dash near heater core. Lots of work to get at.
Heater coolant bypass pipe It would be the return hose for the heater.
No , if you were to bypass the heater core , you would have no heat
The temperature door actuator controls the opening to the heater core. The heater core door regulates the desired temperature in the passenger cabin.
How do I locate the heater blend door actuator in a Ford f150 pickup
Sounds like you have a problem with the air door actuator that switches it from hot to cold. That car shouldn't have a heater control valve, instead it uses an actuator that moves a door which directs the air flow to either flow through the heater core or bypass it. If coolant is circulating through the heater core, and nothing is stopping the fins on the core up, then you have a problem with either the actuator or the door itself. When you switch it from hot to cold do you hear something moving on the passenger side of the dash? If not, it is probably the actuator, or it could even be the control panel.
It is on the centre side of heater against the firewall inside cabin. You must disconnect steering column and remove the entire instrument console and angle it out on the passenger side. Easier than it sounds. Look for video by 'Makuloco' on youtube
The temperature valve actuator is located on the front of the heater case
did the heater core start leaking? anyway you should have no problem with the air if you bypass the heater core.
Upper and lower radiator, inlet and outlet heater core, bypass,Upper and lower radiator, inlet and outlet heater core, bypass,
you would have to bypass the heater core first.
You can buy aftermarket headers with no heater boxes.