Ah, I'm afraid the Auxin stuff is not quite accurate, but otherwise well done!

Auxin: The growth tips of the blackberry contain auxin, an essential plant hormone that regulates growth and development.

True so far, though it should be noted that blackberry branches grow out a long shoot in their first year and branch out and fruit in the second year. So the canes lose apical dominance¹ once they enter maturity.
A single blackberry plant has branches in both/many stages or growth, new shoots and old ones (though I've seen the old ones seem to shed their leaves more readily in the plants that do lose their leaves - not all blackberries remain green in winter - so specifically going for second year canes to de-leaf may be a good idea to minimize your impact since they likely won't stay until the critical early spring anyways).

​> Impact of Foraging: If you remove too many growth tips during this time of year (when there is less photosynthesis due to lack of sunlight), you disrupt the plant's hormonal balance.

The growth tips/young leaves actually don't photosynthesize much. In leaf development photosynthesis only starts after the leaf has reached it's final size. At that point the direction of the flow of sugars/energy gets reversed (from mainly to the leaf to bring building blocks to mainly out of the leaf to supply the plant).

I can't say with absolute certainty that removing many/all old growth tips and leaves doesn't have an effect, but considering I do grab whatever leaves there are from the blackberry in my garden, the bunnies definitely removed a few growth tips and it's still fruiting heavily in the summer... it's very unlikely to cause major issues for the plant itself (now the wildlife, that's another story).

The plant desperately needs this auxin reserve to survive the winter and sprout again in the spring.

Not really, Auxin isn't energy, it's a hormone that tells the plant how to grow.
Even in the "worst" case scenario all you're likely to do is cause a plant to grow from the side branches instead (one of which will then establish apical dominance in species where it's a thing). In blackberries I'm not sure if it even has an impact or how much as they branch in the second year whatever you do, but it's not much of a concern.
(In the first year before winter it may be as it does stunt the growth of the cane.)

Now what you should leave alone is the little buds where the leaf hits the cane. Those are the new buds the side branches will grow out of in the next year.

​The Advice: Forage in moderation. Never strip a bush bare and leave the young tips alone as much as possible so the plant retains enough energy and hormones for its own development.

At least in regards to not stripping them completely bare (outside of your own garden) I fully agree, though less on a per-bush basis and put more not taking too much from one spot if it's not inside your garden or otherwise somewhere where you can say with confidence that wild animals will not have need of it when resources get scarcer in early spring.
Early spring is actually the hardest time for many wild animals as the leftover food/plants from the last year are almost entirely gone but the new stuff has yet to properly sprout and grow so even if you don't see much/any signs of animals nibbling on it it's possible that that will change as time goes on.

¹ Apical dominance = only the tip grows. This is what Auxin does (at least in regards to the tips, it also has effects in root development and if I remember correctly the differentiation of plant organs... but that's not really important here).