There are two things going on.
Cold start mode
For emission reasons, the ICE runs at low RPM producing little power for ~20-30 sec after ICE first starts. This is sometimes engaged after long downhill runs with ICE off. This is a very different, but short-lived, warm-up mode.
Hybrid mode
Once cold start's over, the car balances EV and ICE output, based on HVB charge and your throttle foot. I wonder if you're describing normal hybrid-mode operation. As I recall, there are four hybrid modes. Add in the effect of HVB SOC to change the ICE/EV balance and it covers pretty much everything.
- ICE only
- EV only
- EV + ICE
- EV - ICE (negative split mode)
This last one is a unique case of high MPG with ICE running - negative split mode. It only occurs when HVB SOC is very high, and your throttle foot very light. The "charging" motor acts to slow the ICE RPM, so fuel use falls as mileage soars, but the ICE keeps running. Not common, but neat when it happens.
As to which accelerates faster, EV or ICE, it's a bit of a red herring. In a power-split based hybrid, ICE speed is mostly constant. It operates as a "stationary engine" running any RPM required to generate the requested power. When I start from stop, the engine speed levels off as the car begins to roll, and stays constant until I back off at cruising speed. To anyone accustomed to shifting a manual, constant engine speed is a big change.