Your question reminds me of Grothendieck's Esquisse d'un Programme, specifically the notion of "tame topology". Was this on your mind when posing the question? I am thinking of the part that starts like this:
After some ten years, I would now say, with hindsight, that “general topology” was developed (during the thirties and forties) by analysts and in order to meet the needs of analysis, not for topology per se, i.e. the study of the topological properties of the various geometrical shapes. That the foundations of topology are inadequate is manifest from the very beginning, in the form of “false problems” (at least from the point of view of the topological intuition of shapes) such as the “invariance of domains”, even if the solution to this problem by Brouwer led him to introduce new geometrical ideas. Even now, just as in the heroic times when one anxiously witnessed for the first time curves cheerfully filling squares and cubes, when one tries to do topological geometry in the technical context of topological spaces, one is confronted at each step with spurious difficulties related to wild phenomena.
To me the notion of a "false problem" is a very succinct way of expressing the problems associated with going from (1) to (2) in your formulation, at least if I have understood your question correctly. Namely, in a first course in point-set topology, one spends a lot of time learning about topologist's sine curves, long lines and Sorgenfrey planes, and the problem is that all of these are cooked up to solve/exemplify problems that are "false", i.e. not very natural/interesting to someone who actually does geometry in his day-to-day life.
If we then believe Grothendieck, the problem is not one of didactics at all: the difficulties in passing from (1) to (2) are an inevitable byproduct of the fact that point-set topology is not adequate as a framework for geometry. Maybe this is true. In a first topology course, one does need to spend time on pathological behavior, simply because anyone who does topology should know that a topological space with no extra conditions can be weird and not very geometric. I think this will be true as long as topological spaces are considered the basic framework for topology. Then one does have to unlearn the instinct to think about pathologies when actually doing topology. On the other hand, of course you can't tell students in a first topology course that the pathological spaces they are learning about will not actually be so useful later in life; no one would learn the material after hearing something like that. I suspect that I have just restated what you said in a clumsier manner.