You subscribe to multicast groups, when the cast happens, you have something to receive it, or you don't that's up to you.
In the old days we have these glass tubes, you turn them on and the streams appears on the front, when they're off you can't see the stream.
We had little black box underneath the tube, and if you push the right incantation of button, it would store the stream and you'd watch it later.

I know that might sound a little far fetched, a little magical, but the black square in your pocket that receives emails, it can receives those casts as well.
Depending on how much spam the manufacturer has injected in it, it can probably store around a couple hundred hours of videos, and a couple hundred million tweets or "short text messages" as they used to be called

From the research I've done since posting that, it has become evident that all the little internet fiefdoms that make up the net, each want a slice of the pie, and CDN networks, a parrallel pseudo network to the internet, is the bridge where they get to collect that toll.

If multicast worked as designed, this toll would be in danger, because anyone could just use it instead of the CDN or using unicast and local caches.

No the streams wouldn't be "constantly offering your bandwidth" that's a "broadcast" a broadcast you do always receive it, but a multicast you need to enter the multicast group or else you don't receive it.

How would I not receive streams that I’m not interested in?

That's the same as above, you just don't subscribe or you unsubscribe to the multicast group.

There would be multicast groups just for knowing what streams you could join if you wanted to.

Your street would have a multicast of just your neighbours and just for text.

This might feel foreign to you, but that's because the software to browser is as non-existent and the capability of multicast on the internet itself.

Nobody is going to make these browsers and pseudo internet VHS, when we all know that ISPs will never allow multicast through unless forced to by the statists, finally a legitimate use of force besides building roads and powerlines !

What would decide which broadcast packets do or don’t get sent to my router?

multicast subscription to multicast groups, a trivial affair, it was trivial in 1980s, it is still trivial now, the ISPs will tell you this is a complex scaling impossibility, they are lying