When contemplating whether or not to remove a tree we usually take several considerations under review and then make a suggestion on the value aggregate .
Some questions are : Does the tree help or hinder solar gain in either the summer or winter ?
Does the tree have a root system that impedes growth of other plants that have hierarchy in the planting scheme or usability of the space and or is causing or projected to cause property damage ?
Does the tree provide needed privacy screening ?
Does the tree work with the overall owners maintenance budget - can you afford to keep the tree healthy for its life span if it requires yearly maintenance ?

In regards to requiring a lawn for your child to play on, in a situation like this you might want to research artificial turf. There are a variety of quality faux turfs available, some have earned a bad reputation due to their back fill mix while others have high marks for safety.
Research and proper installation is paramount to choosing the right faux turf for your area.

We've installed quite a few faux turf projects over the years and when you take into consideration all the site conditions and the clients needs, it can be a fantastic outdoor surface choice.

The houzz.com website has a ton of great inspirational photos to look at . Use their search engine to narrow down your search, ex. small play yard, artificial turf.

Here is a link that might be useful: houzz

If it is indeed a maple, you will never get grass to grow--or much else either. Also, you cannot "trim" a large maple successfully. Removal is likely to be quite expensive--I'm thinking several thousand dollars at least by a reputable firm--and as close as it is to other neighbors and your house, you definitely want a licensed and bonded company with plenty of experience.

Since it's all on the north side of your house, the tree is giving you no shade in the heat of summer, so removal is probably your best bet. After it's gone, you could plant a tree that will stay quite small and doesn't have the fibrous surface roots of the maple--perhaps a Service berry, for example.

I agree that the tree will be the first decision hurdle. It looks like a silver maple, but pic is not clear enough to tell conclusively. If it is, they are known to die back in chunks and drop lots of trash. Nevertheless, I don't feel quite as bleak about the prognosis as laceyvail because you can raise the canopy (letting light in so other plants will grow below) and you can commit to regular maintenance, if that's the better option for you. The tree doesn't look that large (based on the trunk) so I don't think removal will be as expensive as predicted. Another possibility with low-rent trees like silver maple is to turn it into a pollarded tree, in order to control its size and keep the wood "forever young."

Based on the tree remaining in place, I'm suggesting that you move the gate to the left (a relatively minor operation) and upgrade the walk. It could incorporate a seating area. Whatever paving material you use, it should be flexible because of the nearby tree roots. Artificial turf, could be one possibility. Another could be mulch, or tamped, granular material. While not classy, I've seen people use discarded carpet as a "paving" material where they don't want to spend any money. It's better than dirt as a temporary solution. I would be nice to paint the fence so it looked fresh. If possible, adding finials to the posts would upgrade the look. A low groundcover would look a lot better than mulch or dirt.

The picture is a generalization, not a prescription. Adjust to suit for likability and workability.

I'm guessing the tree roots will be an obstacle to growing anything. In an ideal world with a large enough budget, I'd remove the tree and put in a sandbox. When outgrown, it could be repurposed to a vegetable or flower planter. I'd paver most of the area -- perfect for a tricycle-- then install a section of those interlocking rubber pads for rough and tumble play. Think playground now, and walled garden in future.

Thanks all! this has really given me some great ideas.

I want to repeat that the roots of maples, the worst being Norway maples and silver maples, make growing ANYTHING nearly impossible. It's not the shade; it's that the fibrous roots suck all water and nutrients out of the soil. And you can't pile on fresh soil or dig a hole in the root system; the roots just fill in. I once managed to get some Liriope spicata (not L. muscari) established near a silver maple, but it took years and the planting was never very impressive.

If you keep the maple, then put down wood chips for mulch in the entire yard and plant a few things in planters.

If one picks a vining type groundcover (that tolerates the given light conditions, of course) it will grow over surface obstructions like roots.