Depending on the tankless unit, most have a variable burner to help keep the temperature where set, and some also have a throttling valve to slow the water if it cannot achieve the desired set point. It sounds like yours is slowing the water flow so it can increase the temperature rise. You may get more flow if you lower the setpoint on the WH. It may change considerably come summer, when the incoming water temperature can rise a lot, making the load on the tankless much easier. This is a common issue with a tankless unit, especially in areas where the ground water temperature can vary a lot to allow it to get quite cold. I've measured my incoming cold water down to 33-degrees F after a cold spell...a tankless system would have to work really hard to achieve decent output levels under those circumstances. Many people only look at the 'nominal' output, but that's based on incoming water being 55-degrees F in most cases...it can go down a lot as it comes in colder...in my case, 22-degrees, which would take your 120-degree setting down to 98 at the same flow rate, or, as you found out, lower the volume considerably. Think moving your hand through a candle flame...do it slowly, you'll burn yourself...do it fast, and you may not notice it at all. The water is likely moving at about 5fps through the heater...not leaving it much time to transfer heat to it along the way.
Some tankless systems can be ganged...i.e., install multiple units to then be able to increase the available volume. That can get really expensive, but if done right, can provide whatever volume you may desire. Otherwise, you learn to live with it!
If you lower the WH set point, you'll likely need to adjust the temperature limit stop in the tub valve to allow less cold mixing. Some valves won't let you get to all hot, so you may need it a bit above your desired mix temp so at least a little cold can be mixed in at the tub.
FWIW, I put in a 6', deep air tub. I thought about a tankless unit, but realized it wouldn't work for me where I live, in the place where it could be installed. Since I have hydronic heating, I just made sure my indirect WH was large enough to fill it without issues...