During my career I have seen aircraft helping the tug with engine power a few times, usually successfully. But it wasn't a good idea then and still isn't, it's too unpredictable.
If the tug couldn't extricate the aircraft then the situation required some careful consideration. The pilot wouldn't have had a complete picture of his situation without getting out and seeing for himself.
I presume in this case if they called for taxi the tug had been disconnected. Looking at the pictures it could be that the reason the tug didn't stop was that it lost traction on ice and couldn't stop before the grass.
I'm paid to consider the hazards of my actions. I have absolutely no idea of the strains inflicted on the struts, pulling out of mud like that. This ain't a Jeep. The gear is being drug through the mud, while holding up the weight of the aircraft and any dynamic forces caused by hitting ruts or the edge of the pavement. Guts got nothing to do with it.
You really have very little understanding of the forces your landing gear is designed to be subjected to. Think of the strain on the struts when you're sitting on the breaks at ref thrust, or how's about the plowing it into the runway at 150 knots. If you're going to worry about anything it's engine FOD.
Moral of the story, logic says give it a go but that's not what they want you to do, so don't. Go have a cup of tea while someone else on big bucks gets to make their decision.
