Actually, I think you mean the president can't make bills. Half the president's job is to make laws by approving bills that have been preprocessed by congress. Regardless, the president has extreme political influence and by lobbying for a bill personally the bill is much more likely to go through. Furthermore, the president has the power of signing an executive order which does not need to be approved by congress and goes into law regardless. So, you're pretty much wrong on all counts.