Harder tamping will not change the flow rate specifically (the coffee expands when wet, and will frequently expand to a size determined by dose weight and grind size - until restricted by hitting the shower screen in the case of 16-20g doses usually.). In fact, especially with off-the-shelf beans, harder tamping is likely making the issue worse by causing the puck to 'crack' and causing channeling (water flowing too fast via holes/streams in the coffee).
Try dosing it level with the top of the basket, do some WDT (or just grab a paper clip, extend it, and do a counter-clockwise stirring motion - center to edge, and move around the basket edge clockwise), and do a medium tamp all in one go (grounds should hopefully end up about 1/2 inch or slightly more, below the top of the basket).
Then pull the shot - however make sure the machine has been on and heated sufficiently before pulling. 15 minutes should do the trick, and that includes the portafilter itself.
Have the new Gaggia Baby ABS myself, and have even tested Lavazza Espresso beans (the black packet) on it
