As always, many, many thanks for your time and advice, @KFrank!



 KFrank:

Because your data is not labeled with probabilities,
you’re not training your model to predict probabilities.


Thank you for the explanation re the above!  That makes it much clearer why my ‘likelihoods’ are NOT likelihoods OR probabilities in any real sense and how interpreting them as such could lead to false impressions.



 KFrank:

Note, when training such a network, you should probably stop the
training when your training loss and validation loss first start to diverge
even if further training were to produce a model whose validation
and/or test loss and accuracy were better. This is because you’re
trying to make your predictions best reflect their own uncertainty,
rather than make them as good as possible, but without a good
measure of their uncertainty.


Great tip and explanation!!!  I’ll be playing around with learning rates and early stops to see how they influence model behavior.  I’m guessing a reduced learning rate will be preferable to aggressive learning with an early stop, but there’s one way to find out… 



 KFrank:

If you want to focus narrowly on predicting the “HomeRec” probability,
then perhaps you should train a pure “HomeRec” vs. everything else
binary classifier.


This will be an interesting test.  I’d like to see how my predictions compare.  Prior to now, I’ve been thinking about my multi-label model as if it were a set of parallel binary classifiers, but that’s clearly a bad analogy.



 KFrank:

if the fraction of “HomeRec” vs.“Not HomeRec”
in your training set is representative of the “HomeRec” fraction in the
“real-world” data to which you’ll be applying your model, then perhaps
you do want your predictions to be so biased.


I think this might be key in my use case.  My dataset is ambiguous to say the least.  It’s possible (and in fact, common) for the exact same input to appear multiple times with different target labels.  E.g. I could have three 50 year old men with hypertension, one who was hospitalized with pneumonia but recovered, one who recovered at home with no complications, and one who died of covid related cardiac arrest without ever being hospitalized.  My input “image” would be the same in all three cases since it’s “calculated” from tabular data, not a true image.  I think this is the key difference between what I’m attempting to do and the traditional image recognition use case.  There really isn’t a right answer.  That’s certainly the reason that I’ve been so hell-bent on interpreting my model results as probabilities.
But back to your insight re bias…  I think I actually do want that bias reflected in my predictions, but if so, I’d be making some strong assumptions regarding the quality, comprehensiveness, and representativeness of my data, i.e. assumptions that seem best avoided.  Either way, I think I’ll do some trial runs with class weights and some with batch normalization to see how they influence my results.  At the very least, that’ll be a valuable training exercise.
At the end of the day, I realize this all might be a fool’s errand.  I’m not sure I’ll ever be able to make statements like “diabetes increases risk by 10% for women aged 40 to 50” as I originally hoped, but I’ve definitely learned a ton about PyTorch, so it’s been worth it either way… 
Thanks again for all of your thoughts and advice!
Cheers,
SEH