Some pavers set flush to the soil would probably do the trick.

Is mulch an option? (seems messy, not great when cutting the grass); some rubber material? gravel/rocks?

Pavers was my thought too. The nice concrete colored square ones.

I don't think mulch is a good option. It is messy, and will get kicked/moved out of place as the gate gets used. And can be more expensive in the long run against something more durable.

If you are concerned about how hard pavers are and the potential of little ones falling and hurting themselves.... you can still set pavers and put an outdoor skid-resistant gym mat on top of it for while they are still small enough for it to be a problem.

With the pics being so close up..the advice given seems right..with a pic from further back showing as wide an area as possible ...other ideas may come to mind...

I'm okay with flush pavers - what do I need to do? kill the grass? put landscape fabric down so no grass/weeds grow in between? any need for sand? then put the pavers down? Looking at my pictures, approximately how many pavers should I get to make it look "right" and what size pavers? Thank you!

To help with the above question, here are pictures of my patio (maybe pavers can complement my patio colors or match) and a picture further away from the gate while in my backyard and a picture in my side yard looking into my backyard through the gate.

Patio and showing how close gate is:

My patio pavers:

Picture taken from backyard further back from gate:

Picture taken from side yard looking into the backyard:

Having a hard time figuring out how big of an area to do the pavers to make sure it doesn't look weird, but want to do the minimum number of pavers.