Fuel system? Yeah, likely.
But Coolant as a cause of the symptoms you described, very unlikely.
You wouldn't be the first person to induce his own intermittent fuel-starvation situation. The truth of your situation might be that the period of time you're imagining as "cooling time" might in actuality be "refilling the carb float bowls time".
Did you know that there's a fuel filter contained within the EX250's OEM fuel line to the carbs? There's a short discussion of this original equipment fuel filter in this maintenance tip over at ninja250.org:
https://faq.ninja250.org/wiki/How_do...fuel_filter%3F. There are a couple of really good pictures of the OEM filter and its location on the carbs.
If the problem doesn't turn out to be a clogged up OEM filter, it might be your add-on filter.
Good luck figuring this out.
And... if you do figure it out don't be "that guy" you described in your post by simply riding off on your happy little way instead of coming back here to document/explain what you found and how you fixed it. Show the forum some love is all I'm saying.