In the context of the satanic campaign of Judaising to change the simple Gospel of Grace not works to which Paul focuses Timothy's courage, the curt advice could be expanded for education, but not done in a short lesson such as these J.McA offerings.
If that were indeed the case (which is supported by the Book of Acts and several of Paul’s letters), then your brief summation was a lot more compassionate towards Timothy than JMacA! You focused upon the outside forces pressing in on the young Pastor, rather than JMacA’s focus on Timothy’s alleged “shortcomings”.
I applaud your analysis, which was succinct, straightforward, and much more sympathetic toward the young pastor!
After all, we’re all in this together, aren’t we? :-)