We made the drive out in my Subaru Crosstrek which has 8.5 inches of clearance and regular street tires. We drove slowly and carefully, averaging ~15 mph just as the page owner says. There are only a few areas which require slowing down to 5 mph. But we did see a lot people cruising at 30+ mph, so it's no wonder that this road has a reputation for flat tires.
Drive carefully and you will be fine. Don't believe the hype. A rental jeep is probably unnecessary.