What about a runner buggy or bike/jogging trailer?
My son is 6, nearly 7 and we've had to get creative as he's a flight risk due to his autism, and I've got to say we hardly get any funny looks to be honest. If people do judge they do well to keep it to themselves.
We've had a huack runner buggy for a few years and for 4-6 year old, it's been fab, downsides for me were that the breaks weren't especially stiff so not great on public transport but we don't really use public transport anyway, but it reclined, had a full hood, and the wheels were big so it did well over multiple terrains without feeling like it was towing. It did need an annual innertube replacement though and every 6 months we had to reinflate the tyres, and finding a pump that fit was a bit of a task. Finding a rain cover was also a bit of a task because they all seemed to leave the feet exposed, which wasn't great when wearing low-tongued shoes or trainers in the rain.
We also have used the bike trailers as they come with a push handle, absolutely a no-go for public transport as they're too big, but they're spacious, have a 360 degree cover so great for poor weather, cosy when you throw in a blanket or 2, and also have all terrain wheels.
We have also tried the festival wagon things, we've had 2, one with just a pull handle - would not recommend unless you live in an entirely flat, well maintained area, and one with a push and pull handle, which was better, but I'd still not recommend. The wheels are small, they get gunged up easily, and it is an additional strain on your back when pushing and pulling. Pros are, you can usually take them in shops quite easily, compared to the bike trailer, but they're still not as easy to navigate as the runner buggy, which also had a spacious undercarriage for shopping and was well distributed gravity-wise if you needed to hang bags on handles with additional hooks.
If it's just for getting from A to B I would recommend the bike trailer, but if it's got to have multifunctionality, then the runner buggy is the best option.