Difficult cold start

Still fighting with this issue.
Car stood a week and I tested fuel "squirt". It is strong. So no fuel evaporation issue.
Manual says that cold start should be done without pressing gas pedal. But this thing would never start that way.
Best effort to get it running after standing few days is pump gas pedal many times and then keep it pressed down when starting. This way it soon tries to run, but shuts down after I stop starting. Pumping and giving it very many long starts keeping pedal pressed it starts at last, if there is enough power in battery.

Somehow it sounds like it does not get enough gas to start, but why?

I am confused what causes this? With warm engine I also need to press gas when starting or it would not start. But it starts instantly.

Could it be so simple as bad starter motor? It sounds fine, but maybe it would benefit about new "mini starter"?

Or maybe there is something wrong with ignition causing bad spark? How can I diagnose that?