Not surprisingly, this has been brought up several times. On the talk page of LST:
(2006) https://www.mediawiki.org/wiki/Extension_talk:Labeled_Section_Transclusion#awareness_of_templates
(2008) https://www.mediawiki.org/wiki/Extension_talk:Labeled_Section_Transclusion#Templates
For <ref>, the similar problem was resolved in T8312
Being able to transclude <section> tags would dramatically upgrade the functionality of LST! It just seems nobody has had the time to look into it.

I requested the missing section tag transclusion feature at https://phabricator.wikimedia.org/T179482

In T39256#1390633, @Gregoryhltnz wrote:

It seems it would need to fully-parse page A, expanding all templates, and then look for the begin/end tags and transclude that content. This would seem to be an expensive operation, so would best be served by a new function, rather than #lst or #lstx, it could be #lstvt for "labelled section transclusion via template". That's a pretty crummy name, but it gets the idea across that for performance reasons I think this would need to be an entirely new function.