I saw a Kickstarter set for an old TV show, that brings up the question ...

How do you find licensing opportunities & contacts?
Is there a website with Intellectual Property for rent?

If I wanted to make a set based on Planet of the Apes, Logan's Run, or Manimal ... where do you even start the process of getting permission?

Just produce the product and start to advertise....Disney will tell you.

On the helpful suggestion side....might want to contact a smaller manufacturer (Monster wax, RR Parks or Unstoppable) and see what their professional suggestions may be.

There are shows where you can find available licenses -- i.e.:

https://www.licensingexpo.com/en/home.html

There are likely others as well. Otherwise you'd need to find the agent for the property you want to license.

Looks like there is also a licensing site:

https://www.licenseglobal.com/

I had to go look what licenses are setting up at the LicensingExpo. Lots of options, but many of them have been done before. Yes, I guess manufacturers have to dive deep into research.