According to the memorial Enomoto Kamajirō (Takeaki) asked the European powers to pass along to the Meiji government, the group of men moved to Ezo because the Tokugawa had been limited to a realm of 700,000 koku (ordered by the government after Edo's surrender, about 10% of Bakufu and hatamoto realm prior to the the restoration), it's not enough to support the 300,000 people who were under the Tokugawa. But they are warriors. So they went to Ezo to help open up the frontier for the Emperor. And as warriors, they could help defend the frontier (against Russia, who Japan had been afraid Russia would move into Ezo since the late Edo). Enomoto requested that, once things were sufficiently in place, an offspring of the Tokugawa whomever the Emperor chooses, should be sent as the ruler of the place. According to the memorial, there was only fighting along the way because the domains attacked them without just letting them go on their way, and they had no choice but to fight back in self defense. Enomoto stresses that everything they were doing was for one, the Emperor, and two for their erstwhile lords the Tokugawa.

So it would seem very clear that the men gathered in Hakodate that winter had no intention of becoming independent from Japan. Rather, they wanted to carve out a new domain for the Tokugawa. It would seem then, at least at the time, the election, which took place just after the Meiji government rejected Enomoto's memorial, which saw Enomoto elected as the group's leader, was only supposed to be temporary, until a suitable Tokugawa heir could be found.

There actually was a possibility of re-establishing the Bakufu during the war in the Northeast. The founding Tokugawa actually had the foresight to realize that one day an anti-Tokugawa faction could take control of Kyōto and the Emperor. To guard against such a possibility, the head priest of one of the Tokugawa temple was always an imperial prince, who would be declared the right sole Emperor should such a thing happen. This was likely thought up following the precedent of the Nanbokuchō. And it was possible that such a move would have likewise plunged Japan into a similar, prolonged civil war. Certainly rumors of such a move reached the ears of foreign diplomats and newspapers. But in the end, perhaps wary of foreign involvement should Japan plunge into a prolonged civil war, Aizu and the Northern Alliance never went that far. While it's impossible to know if that extra legitimacy could've changed the course of the war, the fact of the matter is in two month Aizu and the Northern Alliance were defeated. Those who did not surrender escaped to Ezo. It's only then, having first made sure the Tokugawa had a secure heir (moved to the 700,000 koku Shizuoka domain), did Enomoto and a group of Tokugawa hatamoto escape to join the Northern Alliance forces with the ships of the navy they managed to hold on to. By then the war in the northwest was nearing its end and there was no choice for them but surrender or escape to Hakodate. Without the possibility of the/an Emperor appointing a new Shōgun, there was no way to proclaim a continuation of the Bakufu. They therefore tried to get the second best, but was rebuffed.

I vaguely recall reading that the term 'Ezo Republic' is actually anachronistic

This is correct. At no point in its existence did the forces that gathered at Hakodate ever proclaim themselves the "Republic of Ezo." Enomoto's memorial above referred to themselves as the 徳川脱籍の微臣, meaning roughly "your humble servants who had exiled from the Tokugawa domain." Similarly, in correspondences with Great Britain and France, they referred to themselves, as translated by Jules Brunet, as "Les Kerais Exile's Tokonghava" or just "Les Kerais de Tokongawa" or "Gouvernment Yezo." The name likely came about because the group courted the foreign powers to try to have them mediate with the Meiji government, and presented themselves as liberals who advocated for progress. One such attempt was Brunet's letter which promised the Ezo government would chose its leaders in accordance with the principles of "universal suffrage". This was never carried out, but the election with samurai getting the vote did take place. The election was then noted in a history of the Meiji Restoration written in 1872 Japan, which was then translated by Ernest Satow with the added note that the election "was done in imitation of the practice observed in the United States of America, where these things are settled by the wish of the majority." This was then further changed by the British diplomat Francis Ottiwell Adams who in his The History of Japan directly called the government in Ezo a republic. It was only after the term was popularized in the west was it translated back into Japanese.

中野和典 (2002). "「蝦夷共和国」の顛末-安部公房『阪本武揚』と独立論."
Ravina, Mark. (2017). To Stand with the Nations of the World: Japan's Meiji Restoration in World History. Oxford University Press.