I’m not an Icelander but I do speak Icelandic proficiently and I’ve read the Grœnlendingasaga in Old Icelandic. It’s true to an extent, but it’s an exaggerated claim. As you mentioned, modern editions of the Sagas and Eddas are printed using a standardized orthography which is designed to look like Modern Icelandic, but the actual manuscripts are not comprehensible at all. The orthography is completely different in manuscript: there’s lots of ligatures, vowel length isn’t marked, and there’s no standardized spelling. Scribes wrote things mainly as if they were writing in Latin with a few special tweaks (like þ and ð). Also, modern editions purposefully ignore phonetic details which were present in Old Icelandic but absent in Modern Icelandic: nasal vowels aren’t marked, no distinction is made between /ɛ/ and /e/ or /œ/ and /ø/, and so on. Basically, the answer is “Yes, sort of, only because modern editors have changed the texts’ spelling system to look like the modern language.”
This claim is only taking things like spelling and morphology into account, which are mostly the same between Modern (Standard) Icelandic and modern editions of the Sagas and Eddas. Vocabulary is harder, because a lot of words no longer exist or don’t mean the same thing as they used to, or they reference things which had specific cultural meaning at the time of writing which get “lost in translation” to the modern culture. I don’t think a lot of people know enough about medieval Norse seafaring to know what an “öndvegissúlur” is, for example (a type of wooden pillar in Norse boats). Syntax is even more divergent between Old Icelandic and Modern Icelandic, and it really takes a lot of doing to get used to the older syntactic system which permitted a much greater range of stylistic variation. This is pure speculation, but I think this idea probably came about during the 19th century independence movement, when Icelanders started looking backwards and identifying themselves with the medieval past to establish an independent national character from Denmark, and when interest in Old Icelandic language and literature really started coming to the fore as a matter of national pride and identity.