In short, yes, there was a very live possibility that 'Arianism' would have become the prevalent orthodoxy and 'won' the day.

It's important to nuance this a great deal though. When Arius appears on the scene in the 320s, and challenges Alexander of Alexandria over whether the Son is eternal, it exemplifies two trends of theological thought that have managed to co-exist for some time without resolving their logical conflicts.

Arianism, properly understood, mostly remains confined to Egypt and parts of Caesarea. But what the flashpoint controversy over Arius did, was act as a catalyst. When Arius appeals to bishops across the Greek east, he does so confident in his position that they will be sympathetic to his theology. And he's not wrong.

The council of Nicaea in 325 addresses the split that was beginning to fracture across the church, and decisively rules against Arius. Arianism, as the specific set of teachings of Arius and his followers, never becomes a force again. But it has a second, less obvious consequence - it makes clear that the theology of most of the Greek east is 'arian' in character, in contrast to Egypt and the West (Italy, North Africa, Gaul, Spain). They would not call themselves 'arian', but the fundamental ontological distinction between the Father and a subordinate Son is there.

Depending on the literature you're reading, they often talk about 'arianism' and 'semi-arianism' and 'neo-arianism' throughout the rest of the 4th century. There has been a shift away from that terminology in scholarly work recently, because it's somewhat misleading.

In the period 325 through until 381, the majority of the church would be characterised as non-Nicene. Those eastern bishops, for the most part, did not embrace the theological position that Nicaea rested upon, and they generally had the backing of various emperors. Nicene Trinitarianism was neither the majority, nor was the Nicene creed considered a binding document of faith the way it was later on.

That meant that it very much looked like an 'arian' form of faith would be the dominant form of theology. And, concurrently, it was this non-Nicene form of theology that was evangelised into the migrating germanic tribes of the period (hence, the Goths, Vandals, etc., tended to embrace Arian Christianity. Though what it came to look like in those contexts is very poorly attested).

All this changed rapidly in 379-381 for two reasons. Firstly, on the theological front there were significant arguments and developments that took place, chiefly due to the work of the Cappadocian Fathers (Gregory, Gregory, and Basil) which provided a stronger theological framework for supporting the language of Nicaea, and rejecting the non-Nicene positions. Secondly, on the political front the rise of Theodosius to emperor, who was from the West and had a pro-Nicene theological advisor, quickly shifted the political fortunes of the pro- and non- Nicene groups. The combination of theological, ecclesiological, and political events saw the affirmation of Nicene christianity as orthodox, and the subsequent council of Constantinople in 381 sealed this by re-casting the way that the 325 council was received - as a definitive statement of orthodoxy for the church.

Prior to this, it would be fair to say that throughout the eastern half of the empire, 'arianism' was at least as prevalent, if not dominant, and that had things gone otherwise, things would be otherwise (which is about all one can say about counterfactual history, "if X had been different, Y would have been different too).

Sources:
In my experience, most entry level material on this period and controversy still relies upon older formulations around 'arianism'. The work of Lewis Ayres in Nicaea and its legacy, and K. Anatolios, Retrieving Nicaea and some others has yet to reshape the way the narrative is popularly told. Despite this, any decent textbook or history book covering the 4th century should give you a sense of how dominant the non-Nicene position was throughout the century.