Is mechanics still workable if we drop inertial frames and just add fictitious forces whenever needed?

I am thinking can we throw out first law and concepts of inertial frames completely by allowing for fictitious forces?

Anytime something does not move in a straight line, add a fictious force in the second law. And then analyze the motion as regular. In the particular experiment, we could do some varied experiments to find the fictious force, and then continue.

Would this approach suffice to analysing mechanics or is there somewhere where I would have to be careful?