I'd like to elaborate on what /u/hannahstohelit said in a couple ways. First, I'd like to note that not only were people familiar with the Hebrew alphabet, many were literate in it, because of its use with Jewish languages. A Yiddish or Ladino or Judeo-Arabic speaker would use the Hebrew alphabet for their native language, and perhaps even know spellings of Hebrew words (since to some extent many Jewish languages written in Hebrew retain Hebrew spellings for Hebrew words). In fact, some might've been literate only in the Hebrew alphabet (though this would've been on the decline by the 20th century)--if you speak Yiddish natively and know some Hebrew for ritual purposes you'd probably be literate in Hebrew and Yiddish (both using the Hebrew alphabet), and depending on your profession you could get away with only speaking a Slavic language on a rudimentary level and not even being literate in it, and unless you were an intellectual you wouldn't need to learn a Western European language.

For instance, early 20th century scholar Louis Ginzberg was born in Kovno (modern Kaunas in Lithuania) and lived in Eastern Europe until he was around 20 (EDIT - correction - until he was a teenager), and in his writings he notes that he is not proficient in any Slavic language--not Lithuanian (the language of the countryside), nor Russian (the language of government since Kovno was in Russia then, and the language of a significant proportion of the population of Kovno), nor Polish (the language of a significant proportion of Kovno at the time). Kovno was about 1/3 Jewish, so he could travel easily in the Yiddish-speaking world, and while he may've known rudimentary Polish or Russian or Lithuanian (or some combination) to be able to ask for directions or read a train timetable or buy something in a non-Jewish store, he didn't need to be proficient in them. He needed to know Yiddish, Hebrew, and Aramaic for his religious study, French and German for his academic studies, and several ancient languages for reading ancient primary sources, but not Polish or Russian or Lithuanian. That wasn't true of every person in that position--many Rabbis of that era did know local vernaculars or government languages, and certainly people whose jobs required more interaction outside the Jewish world, but it does speak to the level to which people really could live their lives without much use for any writing system but the Hebrew alphabet.

But the second point is that there was, in fact, a movement to write Hebrew in the Latin alphabet! Well...movement might be overstating it a little. The main progenitor of Modern Hebrew as spoken vernacular was Eliezer Ben-Yehuda. His son, Itamar Ben-Avi, has the honor of being the first native speaker of Hebrew in centuries (which involved borderline-abusive isolation from people who spoke anything else, and a childhood that seems to have been utterly miserable). This gave him a lot more weight in Hebrew-language-opinion than a random person, and he was significant interest in writing Hebrew in Latin characters. He had support from some pretty big names, too--Zeev Jabotinsky (major leader of Revisionist Zionism), Arthur Koestler, Justice Louis Brandeis, Tamar De Sola Pool (president of Hadassah), Meir Dizengoff (mayor of Tel-Aviv), though not all these supported wholesale replacement of Hebrew script with Latin.

There were a couple reasons why people supported using the Latin alphabet. The thinkers who were writing about Zionism and most early immigrants were not the Hebrew-alphabet-only folks I referenced above--they tended to be involved in European intellectual life. Many spoke European languages fluently. Jabotinsky, for example, did speak Yiddish, but he went to Russian-language schools and was a journalist in Russian. So those people would've been comfortable in Latin characters, maybe more than in Hebrew. Jabotinsky in particular found learning Hebrew in square script quite challenging (I'm not sure if he was literate in Yiddish at all).

Another is that Hebrew presents some challenges for people learning it. Hebrew does not indicate vowels in a consistent way--certain vowels are indicated with letters, but those letters do double duty as consonants. Familiarity with Hebrew language, particularly the paradigms for forming words from consonant roots, helps this a lot--but it causes problems when large numbers of people are trying to learn Hebrew. Some letters make two different sounds, and multiple different letters have the same sound (the extent of these depend on pronunciation system). A minor inconvenience people sometimes get huffy about who don't know Hebrew is that some letters have special "final forms", where it has a different character for the last letter in a word.

Itamar Ben-Avi had used Romanizations to teach others Hebrew with success, and had published newspaper columns in it. Eventually he went on to publish a Hebrew newspaper in Latin characters, along with a collection of poems and a biography of his father. The intended readership would've been mostly Western European Jews most comfortable in Latin scripts. He believed that since Hebrew is a Canaanite language, and in ancient times really not so different from Phoenician, and the Latin alphabet is derived from Phoenician (via Greek), it still "counted" as part of the history of Hebrew. After all, Hebrew square script is really not native either--Hebrew adopted the square script from Aramaic, replacing Paleo-Hebrew script (though the similarity between the two is much greater than square script and Latin).

It's worth noting that Ben-Avi was a bit of a westernizer, was was Jabotinsky, and integrating Hebrew into the world of European languages would've been helpful in this regard. It would not have been totally unprecedented--Turkish switched to using the Latin alphabet (Ben-Avi actually claimed to have given Ataturk the idea), and systems that would eventually develop into Pinyin were developing during this time. Ben-Avi also had some pan-national dreams of Jews and Arabs sharing a country and speaking each other's languages, and using the same script would help that in his mind, and Latin would be "neutral" (there was no interest among Arabs in Palestine for such a project, who did not see Latin as neutral).

Despite his stature, this was highly controversial among Jews--especially the biography on his father, but also his weekly newspaper, Ha Savuja Ha Palestini, and later his newspaper Deror. Part of the criticism begins with the title--his system was, in a word, bizarre. 'Ayin can be a few different sounds depending on pronunciation, but none is rendered customarily by "j" in any European language. Several Hebrew letters require multiple Latin characters, making the system clumsy. But more importantly, they represented the trading of the Jewish script, used for Hebrew for centuries and many other Jewish languages, for Latin script. This upset people enough that Itamar Ben-Avi was beaten up by a group of assailants. And honestly the benefits were limited--the main beneficiaries would be people learning Hebrew, and making a text clumsy for a language's few readers would hamstring his efforts. There were Rabbis who were upset that Ben-Avi romanized portions of the Hebrew bible (though some supported the project in general, since it would allow square script to be used for religious texts and newspapers that'd be discarded would be in Latin script). Some opposed it because they would've put pronunciation differences between Jewish communities into writing, rather than allowing writing to be uniform and pronunciation schemes to differ. Also some of the letters where multiple letters make one sound do act differently in different contexts in words, so losing that makes learning the language odd in different ways.

Ultimately his efforts failed. While a circle of intellectuals found the project interesting, most Hebrew speakers found the project somewhere between bizarre and offensive (and embarassing, for the first native Hebrew speaker of Hebrew to advocate this). While potentially useful for people learning the languages, just as many would already read Hebrew script (if they were literate in Yiddish or another Jewish language), or not know Latin script (if they were from a Arabic-speaking country). Most people willing to learn Hebrew were willing to learn the script too. And the complaints about Hebrew script were outweighed by its long history of use.

A similar effort with Turkey succeeded. The Turkish intellectual establishment supported using Latin characters and made significant efforts in that area. Many Turkish-speakers were literate in neither script, so Arabic script had less of a headstart. The difficulties of Turkish in Arabic script were much more acute than in Hebrew, which despite difficulties for people learning it is well-suited to the language.

The proposal got more momentum than it probably deserved because of Itamar Ben-Avi's support. Many thousands of people knew Hebrew in its own script, and millions knew the alphabet even if they couldn't speak Hebrew. While some learners found it helpful, creating a gulf between "normal" Hebrew and "Latin Hebrew" would not have been desirable. Hebrew script was preferable for historical and nationalist reasons, to Romanization never really stood a chance. This went to the extent of even printing music right-to-left to avoid using Latin characters (which has not been replicated, but honestly other systems are worse, not sure why this didn't catch on more).

The short answer- while Hebrew was revived as a VERNACULAR language, it had been in continuous use as both a written and a liturgical language for millennia. It therefore made sense to continue using the same alphabet that not only had been in use in that regard for so long, but was also the alphabet used for Yiddish, which also had been used for centuries. Most people who learned Modern Hebrew already were familiar with the Hebrew alphabet. I wrote more about the history of the Hebrew language, and why it wasn't a dead language that was rediscovered, here.

(There is also an element of the Zionist/Hebrew language movements wanting to be very distinctly Jewish and leaning into Hebrew and its alphabet in that respect, but honestly I think it's a distant second in terms of reasoning. There would never have been any reason for anyone to want to write Modern Hebrew in Latin characters.)