Should I intervene with my daughter's sharing?

My daughter is 3 and the other girl is 7. Here is the situation. My daughter has several riding toys. She has many things to ride on. The seven year old has no riding toys at all. The only toy my daughter has that is big enough for the seven year old to use is her scooter—which also happens to be my daughter’s favorite riding toy.

I was sitting with the 7 year olds mother. We noticed that the 7 year old was grabbing at the scooter and whining “Give it to me!!! NOW!” Then she started whining that my daughter wouldn’t share. Now this may sound horrible, but IMO, the seven year old was acting like a brat. I know she has no riding toys at all, but she didn’t need to be grabbing the scooter and trying to make my daughter fall off so she could take it.

Her mom said nothing. I said I didn’t want to get between the kids and they needed to work it out by themselves. The other mother said “You can do something about it. You’re the parent!” Meaning that she wanted me to force my daughter to share the scooter. I was embarrassed when she said this.

I then tried to take the scooter from my own daughter. I ended up picking her up and walking with the scooter. The 7 year old started grabbing the scooter from ME. I didn’t like that at all. So I took the scooter and put it indoors. I said nobody was getting the scooter. My daughter didn’t care. The girl and her mother went home, and the mother kind of shamed me for not letting her daughter ride the scooter. My daughter said she wanted to hide the scooter whenever the neighbor girl came over.

I feel like this was a train wreck. What could I have done? Explaining to my daughter that the scooter was the only thing the older girl would ride didn’t fly. Also, I didn’t like the way the older girl acted at all. I didn’t like that the mother of the older girl acted like it was perfectly fine for her daughter to try to knock my daughter off the scooter and then threw a fit when I put it inside.

I also think that the parents are kind of presumptuous in thinking that my daughter needs to supply the kids in the neighborhood with riding toys. They could get her a bike or something—used if money is a problem. If the girl had a bike, none of this would have happened.

I like these neighbors but I didn’t like this situation. How could it have gone differently?