The reason English has three different words for those is because English has three different words for 1, 2, and 3. It’s like why we have three different words for sixth, eighth, and twelfth: there’s a suffix here used with regular numbers.
The difference is that instead of ‑th for ordinals, it’s ‑ce for adverbials, and you just aren't recognizing that ‑ce adverbs things — or at least, that it adverbed them once upon a time.
That’s because ‑es was a genitive adverbial suffix in Old English. You can see its remains in all kinds of adverbs that are today spelled with an ‑s at the end.
Note that genitive nouns also end in ‑s. The difference is that making genitive nouns (well, and noun phrases with a clitic ‑’s) is still productive. However, making adverbials this way no longer is so, although now and then people coin new ones along existing models by analogy.
Other ‑s examples of adverbs made like once, twice, thrice include such words as afterwards, backwards, besides, betimes, forwards, hereabouts, needs, nowheres, nights, nowadays, sideways, thereabouts, towards, unawares — with plenty more where those came from. As Janus mentions in comments, there is also a broad set of those which gained a parasitic (=inorganic, non-etymological) final ‑t on top of their ‑es/‑s, such as acrost, against, amidst, amongst, betwixt, gainst, whilst.
Because we made adverbs out of things by adding ‑es (later ‑s) all the time, it was the natural way to make an adverbial out of the numbers. The word nonce has the same origin, although that’s used for a noun not an adverb. In Old English, the word for modern once was ænes or enes, genitive forms coming directly from the Old English word for one, which was án.
By the way, the Old English word án that gave birth to modern one and once also gave us the indefinite articles a, an. But when Old English speakers said ænes (again, that was their word for once), it had two different syllables. That situation was not to last, however, and this led to orthographic changes.
Middle English picked up twice and thrice by the same construction, although with differing spelling and pronunciation than we use today. Then around 1500, as ones became monosyllabic, it began to be spelled -ce to indicate the lack of voicing — and, for the cases of twice and thrice, to indicate a change in character of the preceding vowel.
So what’s going on is that you no longer recognize this fossilized morpheme as meaningful. It means exactly the same thing as those other free morphemes you mention in other tongues.
And no: twice isn't going away. Neither are twin, both, halve, or double. Twain might someday be, though, moving into literary use not bar talk. As for your Italian learners, I strongly advise against saying “two times”; it does not sound right to this native ear. Indeed, the OED specifically reports that twice is:
In all senses now the regular substitute for the phrase two times
On the other hand, tuppence and thruppence are hardly worth a farthing today, eh? :-)
Best Answer
Trimonthly is, in fact, already in several dictionaries. Problem is, unlike its counterparts bimonthly and biweekly, trimonthly is only defined as occurring every three months, sans a secondary three-times-a-month meaning.
I would recommend thrice-monthly, if you needed to form a single word.
As was pointed out in the comment below, the hyphen could be omitted as well.
Also, I'll restate my opening remarks, but add some links:
Although bimonthly can be used either way (see here), that doesn't appear to be the case for trimonthly (see here, here, here, and here). In light of such evidence, I'd be hesitant to use trimonthly when meaning "three times each month."