Yeah, if those tires can turn, they'll burn holes in plastic bumpers. Not too good for chrome bumpers either if they can pick up dirt/grit.
If you plan on doing this often, I'd carry a strap to PULL vehicles as mentioned above. And use the tow hooks provided on most cars. Today's cars have thin plastic over styrofoam for bumper covers--Not made for pushing.
In defense of pushing cars in. The points made regarding how tender the bumpers are on today's cars are valid. As far as pulling cars into the shop I have ran an auto and light truck shop for 35 years.We pull cars into one door as it is facing the street and the drive goes up into the shop. Anytime you are going up or down in elevation you run into trouble with the push bumper grating on the car as one or the other goes up or down. Also, when we pull into our street door we have room to get the pulling vehicle back out.
Now our other door comes off our parking lot into a blind bay. You can not pull a car into it. Also the ground leading into this door is flat, no elevation changes until the car enters the shop at which point the car drops down ten inches. Into this door we always are careful to have enough speed to back off the gas and let the car come in under its own steam.
I belong to a group of professional auto repair shops, the subject often comes up. Many use garden tractors with a push bumper, some dedicated tug vehicles, many now use push bumpers with carpet or tire treads on their four wheelers. Being gentle is the key.
One thing I like about our rig is the spacing between the tires. It allows you to push on the areas between the corners and the license plate. Often the area around the plate protrudes, the area between the plate and the outside corners is generally flat and the best spot to push. We can also turn the blade to maintain even contact during turns.
As for tow points, your dreaming, many cars have no tow point