> Why do the routers involve themselves at the transport layer? Can't they just route IP packets and leave the transport alone?

Because they have to do NAT, at least on IPv4.

> Firewalls -- whose firewalls are we talking about here? If a client (say, home user) tries to initiate an SCTP connection to a server somewhere, what step will fail?

Because the connection tracking that's needed to even recognize whether a packets belongs to an outbound or an inbound connection needs to understand the protocol.

I haven't tried it and am not terribly familiar with SCTP, but from skimming a few references I suspect it would fail when the NAT logic decides it's never heard of protocol number 132 and drops the incoming INIT_ACK on the floor.