What it means
It means that her patients won't be able to consult with her on Monday and on consecutive days after that (the rest of the week). (You guessed correctly.)
With some adjectives or nouns for a time interval or a specific time, English doesn't require a preposition to make them function adverbially. For example:
Janice returns to work next month.
Starting next week, vegan meals will be available in the cafeteria.
Vegan meals will be available in the cafeteria starting next week.
She'll be out of town beginning Monday.
More examples of these kinds of expressions, where a noun phrase for a range of time functions as an adverb, are shown in this answer.
Why??
Notice that the construction with the present participle works very similarly to the absolute construction:
Considering that the conference was over five days ago, it's time for her to return to work.
Her mission accomplished, she returned home.
Journey complete, she returned home.
The conference still in full swing, she could not turn her attention to a new patient's problem.
Provided that she still has the receipt, we will reimburse her for the hotel.
Assuming that her flight was not delayed, she is on her way home now.
Barring unforeseen circumstances, we expect her to return to work tomorrow.
These expressions are very similar to ablative phrases in some other languages, except that English lacks a special noun case for them. I've read that they arose in English by imitation of Latin, where this construction is very common. You are right to notice that it's odd. It's a bit of "foreign" grammar, imported into English the way we've imported foreign vocabulary like kreplach, fiancé, and quantum/quanta, which retain their foreign style.
It's easy to see why beginning on… has remained common and standard even though it works differently from the rest of English grammar: there does not seem to be any other way to express the same meaning that isn't clumsy. None of the three possibilities you tried is grammatical in English, and this is clumsy:
The period during which her patients won't be able to consult with her begins on Monday.
One reason why that's clumsy is because begins is the main verb in that sentence, but the main idea is that her patients won't be able to consult with her. We'd like to make the main verb express the main idea, with the time interval in some sort of modifying phrase or clause. The "absolute" present participle fulfills that role nicely. This might be the best explanation.
