I think you have a great start on this -

Look at IKEA for a rug that is soft but polypropolene / can hose it off on the back deck every summer with woolite. Navy and white sounds great - they have some good options typically. Also, they are inexpensive, so when the kids get a little older you can send that rug to their room and go for something like a kilim here.

I would switch the expedit shelving to under the backpacks /or the wall opposite, because that is probably your best corner for a chair and adjacent table. The bench is so charming I want to put it in front of the fixed side of the slider so you see it when you first come in. That will make a little seating corner. Get a chair with a slipcover - again, IKEA has some terrific options with navy or off-white that you can wash / rotate seasonally.

I think drapery panels for your slider will be best - a wide enough rod that you can pull them open on both sides, that will add color and softness - that way you can let the sun flood in most of the time, but open for light when it is warm and the kids are playing. Plain cream twill curtains with grommet tops can be bought and two panels layered together (fit the first and last grommet over each other) on each side to give you enough width without $$. Trim these yourself with some navy and red grograin print ribbon. I like the idea of straw blinds for the front windows, and you can add the ribbon trim to make it look custom - look at smith+noble for the ones they sell and then diy.

Behind the door, more storage - this is the place for a built-in, with space in the base for the boots / shoe racks now in the front corner. Finally, look for a little kids wood table to put under the art wall with three of their little chairs. Most of the time you can pull that out a bit and just tuck it on the wall when expecting company.

Can't see the last corner (behind the kitchen counter? / island?) but presumably that is the slider end that operates so also the circulation path.

Lakeshore Learning is one of the biggest suppliers of school furniture in the USA. Their items are expensive, but they will last until you have great grandchildren.
www.lakeshorelearning.com

The first photo shows a hinged storage unit that can be closed if desired. If you put one of the short ends against a wall, you will have one square play space with lots of storage, and a different space for play on the opposite side.

The second photo shows a piece of furniture for storing toys in bins, with a play area on top.

The third photo shows a combination water and sand (or cornmeal) table.

The fourth photo shows a table with a washable cover. Good for days when finger paint or making designs on the table in shaving cream is the activity.

You might look at the National Association for the Education of Young Children (NAEYC) site. Good preschools are accredited by this organization. It will give you information about all things three-year old for your young son, all the way through age five, for your older sons.

Best wishes.