Car wants to use engine only 90% of the time

I felt this was the best place to ask because the issue concerns the drivetrain system which is identical to the Prius C. I will explain at the bottom. The car is 6-7 years old and had about 25000 miles when I bought it, I've added about 1000.

Edited to add TLDR: Car uses hybrid system effectively in first 1-5 minutes after cold start, performance very good with motor, then car only uses engine unless very weak throttle, performance bad

Problem:
When I start the car for the first time in a day or after some time has passed, the car very eagerly uses the electric motor and acceleration is good even with light pedal presses. This is regardless of the hybrid battery level. When the battery is low, the engine just keeps running to charge it, but it doesn't affect performance.

When I accelerate hard (but not WOT), the engine's RPMs do not increase, or increase less than 1000 rpm, and the acceleration response is instant like an electric car.

However, after driving for about 1-5 minutes, the car shuts off the electric motor, except for a brief moment when accelerating from a stop, OR when the pedal is pressed very slightly, again regardless of the battery level. This is not enough to maintain typical speeds in my area (speed limits being 30 to 40 mph). Starting from traffic lights also makes the engine rev up to 2.5-3.5k if I want to keep pace and not piss off everyone behind me.

When I have to accelerate hard, such as up hills, the engine spins up to about 3-5k yet it does not feel like I am going any faster which behaves exactly like our previous CVT petrol car.

The power flow diagram in the dashboard display also only shows power flowing from the engine to the wheels and battery and no power from the motor. The only way I can get the motor to spin up is to go WOT which makes both the engine and motor work.

I get between 45 to 60 mpg when I drive carelessly*. I can achieve 70 when I am trying to save fuel and drive like I am 70 years old. The other person who drives this car gets between 40 to 50. In the first state, my fuel consumption seems to be lower as the average goes down, but it happens too briefly for me to measure accurately.
*Okay, by 'carelessly' I mean not being concerned with fuel economy. NOT driving without regard for road safety or other vehicles.

Has anyone experienced something similar with their Prius C, and what could be the cause? This is something that would be very difficult to explain and prove to a mechanic or dealership. Is it just normal operation? I would have thought the electric motor would be activated a lot more. I can almost never use EV mode either as apparently I press the accelerator too hard just keeping up with slow city traffic.
Thanks, from Australia

Why I posted here:
I believe it's safe to assume any experience and knowledge concerning the Prius C's hybrid system shared on this forum will also apply to my car. The car I drive (Japanese Corolla) is almost a Prius C as it shares exactly the same 1.5 hybrid drivetrain, battery and some chassis and suspension parts. Size and weight are almost the same. Even the hubcaps are the same. It is a Japan-only car, so there does not seem to be any dedicated forum, or it would have very little activity anyway.