I don't think there is a technical issue or any kind of complexity at issue here, the problem seems trivial even though I haven't worked the details. It is moot since it's broken on purpose to preserve "They's" business model.
And "They" is the operators of the internet backbone, the CDNs, the ISPs
There protocols for dealing about what you're asking Since multicast is a dead (murdered) technology, I can't tell you exactly what does what but here they are
IGMP (Internet Group Management Protocol)
MLD (Multicast Listener Discovery)
PIM (Protocol Independent Multicast)
DVMRP (Distance Vector Multicast Routing Protocol)
MOSPF (Multicast OSPF)
MSDP (Multicast Source Discovery Protocol)
BSR (Bootstrap Router)
Auto-RP (Automatic Rendezvous Point)
MBGP (Multiprotocol BGP)
MADCAP (Multicast Address Dynamic Client Allocation Protocol)
GLOP Addressing
ALM (Application-Layer Multicast)
AMT (Automatic Multicast Tunneling)
SSMPing
MRD (Multicast Router Discovery)
CBT (Core-Based Trees)
mVPN (Multicast VPN)
There would be many more of course, things that specifically resolve any unexpected issues that might arise from "actually existing multicast" in the hands of "the public", which has never happened.
While I agree that P2P is the next best thing and torrents are pretty awesome, they are unicast and ultimately they waste far more resources, especially intercontinental bandwidth than multicast would.
Also multicast open protocol that might have developed if ISPs didn't ruin multicast for everyone, would have steered the whole internet toward a "standard solution" in the same way that we all use the "same email system". There would be one way that was "the way" of doing this one-to-many communication
To specifically answer your question, as far as routers are concerned, whenever a packet arrives, the router has to decide, WHICH of its WAN ports does the packet need to go to or does the packet need to be dropped.
From the point of view of the router, the whole internet is divided up in the number of WAN port it has and it sends the packet down the port with the shortest path to the destination host.
Multicast is a lot like that, the main difference is that the router MIGHT send the packet to more than one destination.
I think the solution is that receivers that wish to receive the multicast packets to a particular address (and port), from a particular source host, would subscribe to it. The literature mentions "multicast group subscription" I'm pretty sure this is already what this is for.
I think what this does is add branches in the routing table for the subscribed addresses in the multicast range. This tells the router about hosts that become part of the multicast group. I'm not sure if this is supposed to tell every router on the whole internet, or just the routers between the source and the destination host, but it gives the routers that need to know, where to send those packets pretty much in the same way as unicast, except with multiple destinations as specified in the multicast group subscriber list.
It's really just unicast with extra steps, and not that many more steps, and those are all already baked in L3 switches silicon for decades. These protocols were designed to run on computers from the 1980s, I don't believe for a minute that today we can't handle that.