Do network stations not care about being relevant

It's complicated.

Network stations buy an exclusive franchise to network content in an area, so they don't like the idea of you streaming another affiliate.

Also you find different games playing on different stations so if you are a sports fan you might not need NFL Sunday ticket or ESPN if you could stream from anywhere.

It is real fun to watch local TV from far away places because you get to catch network programming in multiple time zones, watch the news in distant cities, etc.