I suppose it depends on the aircraft and pilot as to how dangerous it is to fly with a jacked up pitot tube. We took off our dirt strip in AZ to come back to SoCal and at about 100 feet we went through a swarm of bugs and I’ll be damned if one didn’t fly straight into the pitot tube. We flew 2 hours home with No Airspeed in my 172. It wasn’t that tough and we landed no problem.
Always annoying when some part of your panel becomes inoperative. Typically it is the vacuum pump that goes out, or you are get inaccurate readings because of either a plugged static port or leak in the system. But something got stuck into our pitot system one time and screwed up our airspeed indicator. We reamed it and the tube to it out with a piece of stiff plastic filament and were not able to determine for sure what it was... probably a bee or bug like you had.
We have a Cherokee, several hang gliders and a homebuilt ultralight airplane. Only the Piper Cherokee has an airspeed indicator. These days with even every phone having GPS you can figure out about how fast you are going even if your entire panel has gone dead.
Expensive! It was 30 years old. Never a problem in any air worthiness directive.
I can see helicopter and seaplane (seat of the pants) pilots not being too concerned but I can't imagine a fixed wing IFR plane operating with a compromised system.
Roughly the RPMS and nose attitude should get you by.