I totally agree with your first point, and you lost me on the last sentence. Competition is a great data point, but just because you don't have any competition doesn't mean you aren't solving a real problem.
If there is a demand for something, somebody somewhere has most likely already cobbled together a solution to meet their own need, and then turned around and resold that solution to other people.
It's extremely rare to find a widely spread problem with zero pre-existing solutions or competitors in that field.
Escaping your competition by blowing past them is a fundamentally different concept than finding problems to solve that don't already have solutions and existing competition though...
On the other hand, lots of people are solving problems which are not real.
I agree that finding a unique problem is possible, but it is more probable that you are missing something.