The execute bit is somewhat confused between being a permission and an object type identifier.
And, no you cannot "always copy" the file to your home directory: only if it is readable to you.
Files can be executable to you, but not readable.
You're right in that if a file is readable to you but not executable to you, you can copy it and flip the execute bit and use it. Maybe. But it might not work. The executable may be sensitive to where it is installed. Or the file may depend on its setuid root bit.
I wouldn't design a permission system that way starting from a clean slate; it doesn't entirely make sense. The permission to execute would be separate from an executable type attribute, and execute permission would not be overloaded with search permission (even if it was stored that way; the API would not reveal it at the bitmask level).