For most people most issues would in at their home network. So that's a good first guess for any connectivity problems. Rarely it would be somewhere between your home and the ISP. If it's a small rural ISP then it might be ISP->Internet though I'd think that's rare. Most large scale ISPs have enough redundancy and capacity.
As someone else mentioned ICMP addresses certain classes of failures if enabled but I think the historical reason is more along the lines of the Internet was meant to run over lossy connections. For example, when a certain link is saturated routers will just start dropping packets. Reporting each dropped packet back to the sender is just not a good idea, it adds load to a system already potentially operating at capacity. TCP assumes packets can get lost and retransmits them. When a link goes down routing protocols will potentially send those retransmitted packets over a different link/path. I.e. there's no real concept of "connection down" other than the application layer or TCP eventually giving up (which can take a very long time). The kind of ICMP message that will immediately terminate a connection is when the server machine doesn't have anything listening on the destination port.
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For most people most issues would in at their home network. So that's a good first guess for any connectivity problems. Rarely it would be somewhere between your home and the ISP. If it's a small rural ISP then it might be ISP->Internet though I'd think that's rare. Most large scale ISPs have enough redundancy and capacity.
As someone else mentioned ICMP addresses certain classes of failures if enabled but I think the historical reason is more along the lines of the Internet was meant to run over lossy connections. For example, when a certain link is saturated routers will just start dropping packets. Reporting each dropped packet back to the sender is just not a good idea, it adds load to a system already potentially operating at capacity. TCP assumes packets can get lost and retransmits them. When a link goes down routing protocols will potentially send those retransmitted packets over a different link/path. I.e. there's no real concept of "connection down" other than the application layer or TCP eventually giving up (which can take a very long time). The kind of ICMP message that will immediately terminate a connection is when the server machine doesn't have anything listening on the destination port.