An electric bike sounds like an excellent fit for you. It will be life-changing.
On a mid-priced e-bike, you will beat even world-class cyclists. Here's a perspective on power output to bike wheels:
400 watts - is what Lance Armstrong averaged in peak hours of the Tour d'France
750 watts - what most e-bikes output because this is the max street-legal allowed in the US
2000 watts - peak few second bursts for cyclists in a sprint
3000 watts - what many mountain bike and even some street bikes output
Maybe ride alongside hardcore bicyclists who would otherwise be way out ahead?
At 750 watts you'll beat all of them by leaps and bounds. You'd win the Tour d'France.
Would they make family bike rides easier?
I'd suggest getting a pedal-assist bike instead of one with a throttle like a motorcycle has because it makes you feel like superman when pedaling. Pedal-assist bikes have torque sensors to match you pedaling power to motor output. Higher priced bikes usually have much better torque sensors, and it makes the ride more pleasant.