Place in Loxahatchee, never been there. Google maps says 51 minutes each way. I bet that time is on a good day with no traffic.

The traffic immediately north or south of Haulover adds 15 minutes to anything before you hit a road that actually moves.

As for the walk. The closed lot is like SUPER CONVENIENT. I would say the next lot with the lifeguard building (just south of nude area) is not bad unless you hoof it back to the North lot area. My guess by now is more nude people have moved south with the lot closure to adjust. Might be a good thing as folks have been highly concentrated up north and it thins out.
The sand is firm and easy walking but don't know how much ice and liquid you'll carry.

a lot of the regulars have hybrid balloon wheel carts even at the north lot.

Go to goodwill and get a cheap wagon or stroller and re donate it.