I never had this issue before, don’t count my comment as an expert advice. I am a physicist and always work on my car. Anytime, when my car had problem, I try to understand how it works.
Most AC compressor control by a coil. When car is idle, you can check if it is coil problem.
Most car electronic use 12 V DC. Apply 12 V on the coil, you should hear click sound, if not coil is bad.
If coil is OK, run engine and turn on AC, measure voltage on coil, it should be 12 V DC. Be careful what you do this, because car is running. If both works, compressor should engage when you turn on AC. Check Toyota Highlander repair manual, to make sure what I said is correct before do it.