In the annealed state, alloy toolsteels can still be a bit tough, but not bad when you are using carbide tooling. Water hardening tool steels are about as easy as mild steel.
Using coolant with carbide can be a gamble in that the heating/cooling stress on the carbide can cause it to fail sooner than simply running it dry (hot). In lighter finishing cuts with small diameter tools, coolant is probably okay, for the simple reason that the tool is probably running well below rated speed and is not generating so much heat.
Small tools have small gullets so probably the primary reason for failure is chip loading. You need an air blast to move the chips quickly out of the tool to avoid recutting and chipping the tool whenever it is fully engaged (widthwise) in the cut.
Use the correct coated carbides for milling steels.