First of all, to test the derailleurs travel one of course needs to use either manually pull the cable or just push on the derailleur until it hits the stop. But there's a flaw in assuming travel is the problem. If the OP has the derailleur as far out as possible when the chain is all the way in (small-small) and the cable is as tight as it can be and still allow it to hit the lower stop when shifting down, then total travel is not the problem. The cable is already moving the derailleur as much as it can manage with the levers being used. Neither is the problem that the chainwheel is too far out, as the derailleur is not hitting the stop.

We need to know if derailleur and levers are OE, or if you don't know then what model the levers are. We don't even know if this is friction or indexed system. The levers may not have the travel that the derailleur requires. A less likely cause is far too much cable housing, especially if low quality, which would take up much of the lever movement before actuating the derailleur.