This is an emergency roadside assistance plan. They cover "emergency" situations only.
AAA, Coachnet, etc, would deny your claim also. This isn't about Good Sam being lousy or cheating people. It's about understanding what you paid for.
You did not have need of emergency roadside assistance. You had a planned towing need. That means you pay for it.
The only way they would have paid is if you had waited until you were ready to have it towed, then called Good Sam and requested a tow to your shop. Good Sam would have arranged with a contracted tow company and towed your vehicle. It may have taken 20 minutes. It may have taken 10 hours. But they would have covered the tow.