Maintaining libraries on a live site without Composer
Well, turns out it is possible, at least in terms of getting modules like Webform and Entity Print working.  Both of these require libraries and many guides on installing them are emphatic that you MUST use composer for them to work.

I managed to get both of them working without using Composer:

Regular libraries such as those used by the Webform module can simply be extracted to the /Libraries folder. Simply create the directories specified in the admin/reports/status report after you instal the Webform module and put each library there.

Here is a great guide on manually installing dompdf (to use with modules like Entity Print) in the /Vendors directory: https://www.drupal.org/project/entity_print/issues/3081358#comment-13667896.

So far, at least, not so hard after all...