Monks in Ireland wrote down pagan folklore and mythology. Why didn't this also happen elsewhere in Europe (apart maybe in Iceland)?

I am not a specialist in Ireland, so I will leave aside the question of what makes Irish mythography (recording of myth) unique, and instead focus on the larger question: Why didn't elsewhere in Europe participate in mythography?

Well... for much of Europe, particularly in Western Europe, they did! Greek and Roman traditions had been being written down for centuries at that point, and remained prominent in Latin Christian thought throughout the medieval period. Adam of Bremen and Saxo Grammaticus, in Denmark, both recorded rituals and pre-Christian folklore for Scandinavia, with Saxo dedicating 9 of 16 books of the Gesta Danorum to pre-Christian folklore, albeit a euhemerized version of it. The Nibelungenlied, from about 1200 in the Holy Roman Empire, clearly retains a pre-Christian core, though with heavy influence from chivalric romances. The Old English poems Widsith, Deor, and Beowulf all retain clear pre-Christian stories, and the Venerable Bede describes some pre-Christian traditions (though take his writings with a healthy dose of salt). And then there's the Arabic material, such as Ibn Fadlan, which is at least adjacent to your question, but I do not know enough about what it does and does not preserve to do more than acknowledge its potential relevancy.

In the Frankish kingdoms, much less survives, but we still do retain the foundation myth in the Chronicle of Fredegar and the Liber Historia Francorum, which would count at least partly as pre-Christian folklore, though not in the truly mythological sense.

And then we have the Slavic sources, where we have almost no surviving material before the 19th century. However, the mere existence of these folktales suggests some kind of continuity between them and the rituals described by Saxo in book XIV of Gesta Danorum.

Why did Francia and the Slavic countries not document their folklore? Three possible reasons stand out:

They did and the texts simply don't survive, either due to natural loss or due to being destroyed. We would perhaps expect references to them in other late medieval or early modern manuscripts and collections in this case, which to the best of my knowledge we don't have, but that is hardly guaranteed.
There was nobody with a reason to record the myths. Mythography occurs in specific times, for specific purposes, often due to a cultural break. If nobody who is literate and has access to the oral traditions has a purpose to justify writing them down, it won't happen. For examples of some of these specific purposes, check Writing Down the Myths, ed. Joseph Falaky Nagy, 2013.
the oral traditions were strong enough that there was no need to gather them. Oral traditions can be immensely durable, though the specifics of the stories can change to the point of being unusable for detection of pre-Christian myth. However, this oral composition and transmission can retain very old elements. This appears to be the case for Slavic folklore; Leszek Gardela in a conference paper on his current project Amazons of the North used a 19th century Polish folktale an an illustration into his archaeological work on Slavic-Danish connections for the end of the pre-Christian period. Similarly, Eldar Heide has used a distribution of Scandinavian folklore gathered largely in the 19th century to shed new light on the origins of Loki ("Loki, the Vätte, and the Ash Lad: A Study Combining Old Scandinavian and Late Material," Viking and Medieval Scandinavia 7, 2011). In both of these cases, the oral tradition, at least partly, remained popular well into modernity, so there may not have been the sense of a break as mentioned in option 2.