In the translation I omitted the subject
You can't do that in English. You have to supply a subject.
which - in my opinion - is NOT expressed in the starting (Latin) statement
Well, the verb ending tells you that it's third person singular -- either "he" or "she" or "it".
It can't be "you" or "we", for example.
the subject - according to DL - should have been a SHE (??)
I think you misinterpreted the error message.
It may have told you that "She throws the fish on the table." is a correct solution.
But I doubt that it told you that "she throws the fish on the table" is the (only) correct solution, i.e. that it "must be" or even "should be" a "she".
I disagree. Latin is actually very strict about grammatical gender. In fact there are three: masculine, feminine, and neuter. Whether or not that is reflected in the lesson to which you refer, I cannot say.
Let's also keep in mind that these are grammatical genders, not necessarily biological ones. It would be a further mistake to impose modern cultural viewpoints on those of ancient Rome.
There's a great story in Lucius' The Golden Ass where our hero buys some fish in the market for his supper, but an acquaintance comes along and says he has been overcharged, so the acquaintance throws the fish on the ground and stamps on it. "There" he says, "that's sorted that out."
Why is it marked wrong if I say ‘throws fish’ instead of ‘throws the fish’?
In English, countable common nouns almost always need a determiner before them when they are in the singular -- such as an article ("a", "the") or a possessive ("my", "their") or demonstrative ("this", "that") determiner.
"fish" is a countable common noun.
Thus "he throws fish" cannot refer to just one fish -- "fish" cannot be in the singular here, since there is no determiner before it. It has to be plural.
But the Latin sentence uses piscem (singular) and not pisces (plural), and so "fish" is not an appropriate translation here.
It has to be "a fish" or "the fish" (singular).
The fact that english is missing a third person singular form of they is a problem, but I feel that a language should be allowed to change according to requirements. We can repurpose an existing word or we can create a new one. But there is no logical argument for not having one.
I understand, but disagree. English has held its own for centuries, if not millennia, with he/she/it for singular third person pronouns. Why would it need to add "they" - especially since "they" is already used for third person plural? Therefore, contrary to your statement, I see no logical argument to support using "they" as third person singular.
You're absolutely correct that appealing to tradition is not logical. It is fortunate, therefore, that I am not basing any logical arguments on that fallacious foundation. I have pointed out what already exists in English.
What are you trying to accomplish by introducing a singular "they"? Why can't you use the he/she/it that already exist? What is the problem you're trying to solve?
Couldn't reply to your last post for some reason. The short version of all this is thus: We need to use something, and all the current options are wrong. So invent something new, or repurpose something existing. They is the closest fit if you choose the latter, and its what I've been doing for a good number of years.