1. In British English, and I'm pretty sure US English as well, one of the uses of 'would' is indeed as an equivalent of 'used to', that is, to talk about a past habitual action. A bit old-fashioned but definitely not obsolete or rare.
When I lived in London, on Sundays I would often walk near the river Thames.
When I was a child, in winter my mother would buy extra firewood.
2. You ask if you could fill the gap with "we compete". If you did this you would change the meaning from a past habitual action to a present one.
During our summers at the beach, we ...... another to see who could collect the most shells.
The correct way to fill the gap using 'would' is:
During our summers at the beach, we would compete with one
another to see who could collect the most shells.
Note: "one another" is a formal old fashioned way of saying "each other".