You can, but (at least in Europe) it is rarely done.

A big problem is that it doesn't count as a daily rest if the vehicle is moving. This means in practice that the longest you can drive in pairs is 4.5h A, 4.5h B, 4.5h A, 4.5h B, 1h A, 1h B, followed by at least 9 hours downtime. Still a 20-hour non-stop drive, but not even remotely 24/7. You'd need additional standby drivers at a remote base, at which point you could also just swap the trailer to another truck.

In practice this means it is rarely worth it. Combine it with the schedule going out-of-sync with the days, and for regular freight it is almost certainly more expensive than just giving each driver their own truck.

The exception to the rule is high-priority cargo which for one reason or another can't be done via air. Take Formula 1, for example: with back-to-back races they have just three days to tear everything down, drive it across Europe, and build it back up again. It's a 15-hour drive without any delays, so having a solo driver rest for an additional 9 hours halfway through is unacceptable. With a driver pair it is easily achievable, and the additional costs are irrelevant.