Your example (a):
The milk was so hot that I couldn't drink [it].
would be valid if phrased as:
(a1) The milk was so hot that I couldn't drink.
However, either is really short for:
(a1) The milk was so hot that I couldn't drink the milk.
When "drink" is used as an intransitive verb, the sentence does not explicitly indicate what is being drunk. Sometimes it is unclear or unknown, and he speaker may not care; sometimes it is implied.
Let's look at your examples:
(b) The baby still drinks from a bottle.
Here the baby may be drinking milk, or water, or juice, or something else. We don't know, and the speaker doesn't care.
(c) He opened the can and drank thirstily.
It may be soda or beer or something else. Previous context may have made this clear, but for the purposes of this sentence, we don't know or care.
(d) She filled the glass and drank.
This is the same situation.
Now let's look at some other examples:
The animals came down to the waterhole to drink.
Here it is clearly implies that the animals drank water, so any object would be redundant.
(e) Mary takes great pleasure in eating and drinking.
Here any and every kind of beverage is meant, so no object is needed.
(f) John has quit drinking.
Here the clear implication is "drinking alcoholic beverages" so an object would again be redundant.
(g) I'll drink to that!
Here the meaning of "drink" is different, it means offer or join in a toast. Here the object is almost surely "an alcoholic beverage", and neither speaker nor reader cares which one, particularly if the drinking is only metaphorical. Indeed "drink to" could be thought of as a phrasal verb related to but not the same as "drink".
The above examples are largely taken from or modified from these two dictionary definitions:
Similar examples could easily be found in other dictionaries or elsewhere.
Conclusion
It would seem that a good rule of thumb (not an absolute rule) is that the verb "drink" may be used without an object when either any plausible liquid is meant, or other contest has already made it clear what liquid is meant, so an explicit object would be redundant
As I wrote above, I think (a1) with the final "it" is perfectly acceptable, and it does not sound "weird" to me. However, I am confident that the form with "it" is much more common. Why? I am, not sure. Perhaps because the implied object "milk" is separated from the verb by an number of words, and is in a different clause. Perhaps because milk is less likely to be hot than coffee, tea, or soup.
(h) The tea was so hot that I couldn't drink.
The sentence (h) seems more plausible to me than (a1) does, but even here an "it" would often be added. I cant specify any wider principle that explains this.
In a comment, user Maciej Stachowski suggests that the ending of (a1) "...that I couldn't drink." can easily be read to mean "I was unable to drink anything at all. Indeed I can imagine a context in which (a1) took this sense. For example:
(i) I gulped the heated milk. It seared my throat. I tried to ease it with some cold water but i couldn't. The milk was so hot that I couldn't drink.
So this would suggest that the use of the "it" is preferred, that (a) is preferred to (a1), because it avoids the ambiguity demonstrated by (i). This is in line with my rule of thumb because it means that here the listener does care what liquid is meant.
Or, as Maciej Stachowski put it, (a1) is a valid sentence with a completely different meaning, nd the "it" in (a) is needed to rule this out.