"By the way, you keep mentioning how the 'burden is placed on the user of the library', and that's exactly right. The library author cannot and should not be expected to predict everything that can happen. The user of the library is the only one with enough information."
While I concede that this is true in general, that also does not mean that the author of a library should not be permitted to provide sensible defaults to allow someone to use the library with a minimum amount of fuss, and at the same time give the user the power to override these defaults when required.
"You talked about a system using many third-party libraries. Let's look at how that plays out. Because there's so many libraries being used, there's a pretty good chance of conflict. In your system, that would mean that when a conflict occurs, you need to go in and change all the uses of the variable to use the prefix."
In my system, every variable in a separate module in general has to have a prefix, just as Python does. That prefix is set by the library author but can be changed by the user of the library if required. You can dispense with the prefixes temporarily by using import in order to manage this verbosity.
You mentioned knowing 'whether a variable is from the math module or not', but ironically, hyper-static scope handles that case wonderfully well. Because in hyper-static scope, all variables are resolved at compile-time to a unique box.
Doesn't help. Your program knows, but you, the programmer, can't easily know this by mere inspection of the code or a snippet of code. You may even need to compare source files from different libraries in order to resolve this question fully under the system you propose, or use a special-purpose IDE or tools. I think that is much more important. With my proposed system, if you see Math::sin or (import Math ... (sin x) ...) then you'd know where it's coming from. An import form has only local effects that end at the closing parenthesis. Use of hyper-static scope in the way you propose on the other hand has unpredictable global effects that can be difficult to trace. I personally don't feel that it is too much of a high price to pay in verbosity and flexibility, and both of these can be ameliorated to a certain degree by using import.
Don't get me started on IDEs. If a language needs a special-purpose IDE in order to be usable, well, I consider that a very serious shortcoming. That's just another kind of forcing users to do things they don't want to do.