I think there is a missing piece to this conversation, which is about the definition of namespaces. It's easy to conflate "problems with semver" and "problems with semver+the way my package manager handles versioning".
In npm, or with any package manager that allows multiple independent versions to coexist, the version becomes part of the namespace. Multiple different versions of a package can be installed at the same time and referenced from each other. It's no problem if you depend on B1.0 and C1.0, and they depend on D1.0 and D2.0 respectively; you just get all four installed.
In Rich Hickey's vision, we would move the compatibility contract into the package name, so if we want to make a new version 2 of helloworld that does not obey the contract of version 1, we call it helloworld2. In practical terms, this is no different from the above, it's just that we add the version number to the namespace through the package name rather than the package version.
Not surprisingly, this latter strategy is already used by most systems where versions aren't part of the namespace. Debian has python2 and python3. Python has urllib and urllib2 (and urllib3). Unix has mmap and mmap2. It works fine, but it's a bit of a hack.
If your version is part of the package namespace, semver works fine and solves the problem it was intended to solve. If, like in Java, every version has to share the same name, then I agree that semver doesn't really buy you that much. After all, it doesn't help you much to know that your unresolvable dependency tree is semantic.
It's not a coincidence that npm handles transitive dependencies in the way it does; it's the whole point. I'm sure it makes npm much more unwieldy to allow multiple different package versions, but having used it it is exactly one of those "solves a problem you didn't realise you had" moments.
With that said, I definitely agree that more careful API design would allow far fewer backwards-incompatible (ie semver-major) changes, and more tightly defining compatibility (like with this requires/provides concept) is a good way to do that. Many web APIs already say things like "we will return an object with at least these fields, and maybe more if we feel like it", which I think is a good example to follow.
As far as changing package names, there's also an interesting question, not about how computers deal with those package names, but how we do. After all, why helloworld and helloworld2? Why not give it another name entirely? Part of that is because you want to make the connection between these two packages explicit. But if you change your package dramatically, perhaps that connection is not doing a service to your users.
One of the biggest complaints about major rewrites is "why does it have to be the new way? I liked it the old way!" If you genuinely believe the new way is better, why not let it compete on an even playing field under its own name? (And, if you know your new package can't compete without backwards compatibility, maybe don't make a new package.)