It is your road and not the trucking companies'. If possible, they should take part in the process, but it is a DOT responsibility to focus on quick clearance by any reasonable means. Just to be clear, it is DOT not the police who should have the say so on clearance. Once the lanes are cleared, the trucking company can assume responsibility for remaining clearance needs as long as it won't cause significant traffic impact. Use your equipment, towing equipment or contracted equipment, but, don't wait.
Delays are too costly and can be avoided.
Be part of the decision making process on scene for you are the only agency that represents the interests of the motorist caught in the queue. If you are told it is working well, have someone evaluate the records and make sure. Sometimes certain individuals get it done quickly and we think everyone is doing the same. I know of areas where serious incidents may be cleared by one key responder in 60 to 90 minutes while others will take 5 to 7 hours. Just as an example, total station systems have been credited with reducing the time it takes to investigate fatal or criminal crashes.