There's no way cutting down a tree, shipping it and processing it requires a tree-worth of fuel.

Let's fact check that:

(TLDR: it seems you're right)

(Note on gross vehicle weights. I found everything from 80,000 kg at the high end in Canada and 80,000 pounds consistently in the US. That wide range, especially the huge difference between Canada and USA, makes me somewhat suspicious of the following calculations. But I think it still works out in favour of the original assertion.)

1 m³ of diesel contains 38.68 GJ of energy.. That means 38.68 MJ/litre.

Air dried logs (20%) moisture have 14.7 GJ per tonne. That's 14.7 MJ/kg.

Ballpark empty weight of a semi truck and trailer is 35,000 pounds (15,876 kg).

Canadian maximum weight limit for semis is 80,000 kg. I've seen numbers varying from about 35,000 kg to 80,000 kg, depending on jurisdiction, trailer configuration, and permits, so I'll use the biggest number.

That leaves an estimated payload of about 64,000 kg. That amount of wood contains about 940,800 MJ of energy.

Average fuel efficiency is 39.5 L/100 km as of 1999. That's 1,527.86 MJ.

The payload contains over 600 times the energy required to haul it 100 km. Obviously, there are considerations of actual distance, round trips, logging equipment hauling and use, etc. The numbers can change dramatically based on actual payloads, too, but it seems the lumber has "energy to spare" so to speak.