what I have done in the past, is using a twin screw boat, cruise out to the broken down vessel, then reverse up to it gently, once contact is made, give it full power astern and push the boat in. a flat stern boat like that of the wyreforce, Al Khubar, Smit nedland or Aziz tug is best. if you have a pusher tug, just drive past the broken down boat, turn around and come at the broken down boat, and drive upto it, once you have made contact, keep on driving and push it ashore.

as for "burning motors out", pushing a striken boat is no different that having a pusher tug with 3 or 4 barges in tow, as long as the props can spin freely you will not do any damage to the motors. its prolonged stalling of motors that does the damage.

I have used my RTR Southampton:
sail out to, then reverse to farside of casualty boat, and keep on reversing until boat is ashore, using differential thrust to steer it in,

Sydney Star: sail out to casualty boat and pull along side, engage the bow and stern thrusters, then crab ashore, pushing the striken boat. (works ok with small RTR toys, not so well on bigger boats due to power availability.

even my najade speedboat was used once, to collect a yacht, (my yacht) that was done by moving out to it, then turning around then going between the forestay wire and the foresail, once snagged, the najade was brought to the bank, riding the yacht!

a pvc X frame with pool noodles is a good bet, simple to make and works with (PRACTICALLY) any boat as long as the bow isn't too sporty, if it is, the frame can be adapted to be tied to the propulsive rescue boat.

Even something as simple as a box-hull with two propellers, utilizing tank steering will work