When was Football at the Olympics considered important?

Uruguay have 4 stars on their shirt, 2 for their Olympic wins and 2 for their World Cup wins.
Hungary’s Golden Generation won Gold at the 1952 Olympics and then went on to finish runners-up in the 1952 World Cup

During what period was the Olympics considered meaningful and important?
When were teams allowed to send first team squads, and did most teams send first team squads during the period it was allowed?

Was it that Uruguay and Hungary were allowed to send their winning teams of...

When was Football at the Olympics considered important?