I figure this part of Tesla Roadside Assistance was in SoCal:
"WHAT IS COVERED?
Flat Tires In some markets, our trained roadside assistance providers carry a limited number of loaner wheels to quickly replace a damaged wheel or tire. Service Centers may repair or replace your damaged tire at your cost. Please make arrangements with your Service Center for this service. Pricing and availability is subject to change based on location. Loaner wheels must be returned to the Service Center within three days or an agreed upon time with service which will be exchanged for your original wheel. "
So in some markets, a Tesla van comes by with a loaner tire and wheel, not a sub contracted tow truck.
I am not sure where you read "tesla van" in that. It says in some markets "trained roadside assistance providers". Those third parties take the wheel to a tesla SC (thats part of the agreement you sign when you get a loaner wheel).
I have had to do it 2 separate times (including getting a loaner wheel) so I am unfortunately familiar with the process. The third party provider has (or doesnt have, as per their agreement and availability) loaner wheels. The rim comes all marked up with red or stickers, to prevent the person from simply keeping it.
They (can) take your wheel to the service center of your choice for replacement. You sign an agreement to pick it up from the service center within 3 days. You can also take your wheel with you if you prefer to "deal with" the damaged wheel yourself. In both cases, you have 3 days to take the loaner wheel to a service center.
Can you tell me where in the statement above you are seeing "tesla van" comes, or are you simply making an assumption because it says "please make arrangements with your service center for replacement? If the latter, as I said, I can definitively say that their third party towtruck services provide that service at least in some cases, and I have not heard a single case of "tesla van performs roadaside assistance".