It seems to me that almost all major sources of Norse mythology, as well as a lot of sources of history in general, are from icelandic authors. There don't seem to be many manuscripts from Norway, Denmark, Sweden etc. What are some of the reasons for this?
In short, the prominence of Icelandic stuff in Old Norse ancient lore and history was mainly determined in the social prestige and 'utility' given to vernacular poems of the Icelandic elites both out of and within Iceland in the 12th and 13th century, and its integration with literacy and elite culture as well.
There is, in fact, one more major source of Old Norse mythology, called Gesta Danorum ('the Deeds of the Danes'), and Saxo Grammaticus, a Danish clergy, wrote this historical writing little earlier (about 1200 CE) than famous Snorri Sturluson, but in Latin (You can find the dated English translation (in part) here in Project Gutenberg).
Recent scholars surmise that pre-Christian past and its tradition played an important role in both Snorri's and Saxo's writings, but neither of them were blind adherents of the old faith itself (Lassen in Jørgensen et al. 2010; Mortensen 2018: 65-70). In short, pre-Christian past and its traditions were their 'cultural capital', overcome past, on which they build their present identity.
Neither had other Scandinavians like the Norwegians forgotten their pagan past and its tradition completely by then, the beginning of the 13th century, though it does not mean that some of them adhered to their old tradition secretly (as for the trace of possible almost last remnant of pre-Christian community in Scandinavian mainland, it might also be worth checking my previous post in: Chances of Nordic pagans in 13th-century Svealand?).
To give another example, the contemporary historical writing, Böglunga saga ('the saga of the Baglers' - a faction of medieval Norwegian Civil War), though probably written also by the Icelanders and extant also in the Icelandic manuscript, record an episode of the blacksmith in southern inland Norway (Næs in Telemark) and strange traveling 'guest' who asked the blacksmith to repair horseshoes of his horse - later this stranger revealed himself as 'I'm Óðinn, and I took a visit in Norway on my way to Sweden' (ÍF XXXI: 49, 124). The scribe relates further that the fourth day after this stranger's visit there was actually a battle between two rulers (throne contenders) in Sweden (Battle of Lena, on Jan. 31, 1208 CE).
Nevertheless, Icelanders and their poems were highly prized as possible sources for the past among non-Icelandic Scandinavian authors during the high Middle Ages. The following passage is an example:
'Nor may the pains of the men of Thule [Icelanders] be blotted in oblivion; for though they lack all that can foster luxury (so naturally barren is the soil), yet they make up for their neediness by their wit, by keeping continually every observance of soberness, and devoting every instant of their lives to perfecting our knowledge of the deeds of foreigners. Indeed, they account it a delight to learn and to consign to remembrance the history of all nations, deeming it as great a glory to set forth the excellences of others as to display their own. Their stores, which are stocked with attestations of historical events, I have examined somewhat closely, and have woven together no small portion of the present work by following their narrative, not despising the judgment of men whom I know to be so well versed in the knowledge of antiquity (Saxo, Prologue of the Deeds of the Danes: the translation is taken from Elton's one found in Project Guthenberg).
So, what distinguishes 12th and 13th century Iceland from other part of the Scandinavia was the continuing social importance of vernacular (Skaldic) poetry for their elites (Cf. Nordal 2001), and in addition to pedigrees, these poems, partly based on the mythical lore to understand its highly complicated metaphor expression technique known as kennings, were Iceland's cultural treasure of the knowledge of the past. Snorri composed his Prose Edda not as a secret treatise of the forbidden religion, but as a kind of a handbook for skaldic poets, 'Old Norse deities and their deeds for dummy' or something like that. This is the current academic consensus for Snorri's work, and if we accept this hypothesis, the question (s) for OP should be 'why were high medieval Icelanders so eager not only to keep, but also to write their oral traditions of the past down, in contrast to other Scandinavians?', as well as 'Among the oral tradition of the past, why the poetry that presupposed some knowledge of myths were the most important?'
As for the first question, some distinct characteristics of early vernacular literacy in the 12th century Iceland might be a key to understand their uniqueness among the Scandinavians. In contrast to Scandinavian Peninsula where the archbishop's cathedral chapter and its school was the main cultural centers (Cf. Mortensen 2000), two Icelandic bishoprics, Skálholt in South and Hólar in North, and their attached cathedral schools did not monopolize the literary activity in high medieval Iceland. Private church centers like Reykholt (bought by Snorri) and Benedictine monasteries like Þingeyjar, often dominated by Icelandic local elites, also played an important role (Jón Viðar Sigurðsson in Imsen red. 2005: 190-96). They had been founded earlier under the Anglo-Saxon influence where vernacular literacy was flourished. In Scandinavian Peninsula (as well as Denmark), the majority of [a small amount of] previous (Benedictine) monasteries was replaced or shadowed by the reformed monastic order's foundations like Cistercians later in the middle of the 12th century, but in Iceland, the early penetration of the church reform did not largely occur. Thus, de-centralized and locally tied ecclesiastical institutions could produce literary works representing the interest of the local elites in Iceland, in a vernacular language like Old Norse-Icelandic.
On the other hand, concerning the latter question (as well as a part of the former one as well), /u/Platypuskeeper and I approach this problem from different points of view before in: Why did Snorri write the Prose Edda?, so if you have some time, please also check the linked previous question thread.
References:
+++
[Edited]: fixes typos.