I can imagine someone writing a paper using differential equations, but not about differential equations. But I think that the knowledge of this stuff starts on the theory, not on the actual implementation of them. I think it is totally fine if someone asks on discourse the best ODE algorithm given some statement on the problem structure, and then has only minimal understanding of the details of the algorithm (except enough to know where its tradeoffs lie).
If they need more training, it is that people need more linear algebra (and basic numerical analysis) training if they are going to be using differential equations as a central part of their work. if they are doing nonstandard models and have stiff equations, maybe training on conditioning/etc. (see https://julia.quantecon.org/tools_and_techniques/iterative_methods_sparsity.html for example). They also need to have a mental model in order to form heuristics on the right algorithms in the right circumstances. But I am not sure if there are any analogies to that in the standard DSGE, linearized perturbation approaches unless you are trying to do something nonstandard. There aren’t a lot of variations in the standard algorithms where tradeoffs are required.
I completely understand what you mean, and agree with you. I am just saying that you can’t blame Dynare for this 
And to me the worst of all worlds is people rewriting code that someone else has already tested. I would rather they use Dynare and learn more economics (or maybe about eigenvalues and/or fixed points in function spaces if they have limited knowledge).
I agree completely with you on all of this stuff except the conclusions. To me, the failure is that people are not learning enough linear algebra, not that they are spending too little time programming standard algorithms. If you understand saddle-path equilibria, what a perturbation method is actually doing, BK conditions, etc. then I think a student can figure out how to convert between Klein and SGU canonical forms/etc. or whatever if they really needed to. Otherwise, let them use dynare for implementation where it works, and quiz them on the theory (eg. make them log-linearize on paper and do a simple paper example where the BK conditions fail). Global methods are a different animal, of course, and for that they need to actually program things up themselves because no two models are the same.
That you have every right to disagree with, and one I feel less confident about (as it is not my area of research, and is an economics rather than a software one) My point is that dynare is so easy to use that it can lead people to layer on all sorts of moving parts in a model which makes interpretation very difficult. I am thinking of the examples with dozens of variables, shocks, parameters, etc… If you poke the DSGE solution (i.e. it is the properties of the solution which is a black box, not the code or algorithm itself) with an impulse or a change in parameters, mortals like me have trouble interpreting the output. For simple models, this rarely applies, of course. But this is a methodological discussion where I am somewhat ignorant, and could be completely wrong. But narrowly on the Dynare question, I don’t blame dynare since I think it does what it can to generate output for interpretation. At least with Dynare I am confident that any odd behavior is due to the solution itself, rather than being some software bug the researcher didn’t see.