recovery guy was probably worried about further damaging your house and/or car. Hard to tell from the pic, but if you can rope it up from the front suspension, rotate it a bit and pull it out with someone turning the wheel maybe.
If it were me, id make the guy get a professional recovery. If you let them do it DIY stylee, you will no doubt take part as well. If it goes pears you will have to accept some responsibility. Depends what level of DIY the guy turns up with lol, if its a transit van and some rubber band rope from halfords id worry lol!
I've done a bunch of recoveries off road in landies with winches and lots can go wrong, and thats with properly rated kit and proper attachment points. Its hard to judge the forces being applied, and if the equipment is crap its easy to snap stuff. You dont want to be near a snapping rope with a hook/shackle attached!