The problem with this is question is that it all comes down to how you define "relevant". What is very relevant to one person is not the same as another, and news orgs are in the business of sending out as many updates as they can.
I follow the AP directly, which IMO is pretty good about being concise, relatively impartial and generally does a good job of covering national news. You might try a few different aggregation newsletters to see what feels right for you. But you're always going to make tradeoffs between missing stories and receiving low-value updates, because your low-value updates may be someone else's hugely relevant story.
I think “news service that only alerts me about important stuff” is sufficiently specific in that you could probably come up with a service that generally satisfies most people. I think the reason it doesn’t exist isn’t that we don’t know what people think is important, it’s that important news is bursty.
It would have to be fully subscription supported, since advertisers probably want to see an even amount of engagement each week (even if it’s small) so that they can set predictable budgets. But important news doesn’t work like that, sometimes there are 3 huge news events happening at the same time, sometimes there’s a slow month with no really big news happening.
I follow the AP directly, which IMO is pretty good about being concise, relatively impartial and generally does a good job of covering national news. You might try a few different aggregation newsletters to see what feels right for you. But you're always going to make tradeoffs between missing stories and receiving low-value updates, because your low-value updates may be someone else's hugely relevant story.