The commission basically represents the governments of the individual member countries. Because it has the most power by far of all the european institutions, this means that the governments in turn can exercise power at the EU level in a fairly unimpeded fashion.
If the legislative power were transferred away from the commission to the parliament (where it should be, if EU governance were to follow the standard approach for separation of powers), the individual member states would lose a lot of control. Control they aren't at all interested in relinquishing, as you may imagine.
And that's before even considering making the executive branch (the commission) directly elected.