I've found that most Scandinavians (or at least Danes) have a pretty easy time with English.
I think it works the other way round, too, though there's an asymmetry in the fact that, whilst most Scandinavian speakers encounter English regularly, the reverse isn't true.
Nonetheless, even though I've never studied any of the Scandinavian languages, nor ever been there, I find it fairly easy to read written Danish or Norwegian. I went to a Christmas market at the local Norwegian church last weekend, and amused myself by reading all the Norwegian signs and inscriptions. With so many cognate words and an almost one-to-one mapping of word order, I could easily guess the odd word that wasn't immediately obvious.
(None of that either supports or refutes the Scandinavian hypothesis, of course. It's just an observation that there are significant common features and vocabulary.)
As an American living in Denmark for two years now, I would agree that that's true to some extent for written Danish, although it also helps that Danish grammar is just not that complex. But from a linguistic perspective, spoken language is often taken to be primary, and spoken Danish is not very close to mutually intelligible, even if you take a very generous view of it. It's nearly impossible for me to even parse spoken utterances into syllables, or produce most of the phonemes, despite some time attempting, because the phonology is just so dissimilar to English phonology. I find the Romance languages generally easier to approach from a phonological perspective (e.g. I can produce and transcribe Italian sounds vaguely competently, even if my Italian grammar and vocabulary is poor).
You're right that the spoken language has a greater claim to primacy than the written - there are enough examples of diglossia around the world.
However, I think you overstate the significance of phonology. Many native English speakers struggle to understand Glaswegians; partly, that's lexical, but mostly, it's phonological. It doesn't mean that it's hard, though, just that the listener is unfamiliar. An American dropped in Glasgow would soon learn to understand the locals, I feel sure.
You say that you've lived in Denmark for two years but still struggle to parse spoken Danish. That seems like slow progress: would it be fair to assume that you spend your time mostly in an English-speaking environment? I ask because I'm trying to work out to what extent Danish pronunciation is hard, and to what extent you just haven't had much exposure.
Danes tolerance for Danish pronunciation is way tighter than Americans for English, so if a sound is just slightly off, we don't understand a word of what you're saying.
Mixture of both, I think. The fact that Danes speak quite good English, and don't expect foreigners to speak Danish, means that they switch to English as soon as they realize you aren't Danish, so it's quite easy to get by in English. But I've spent some time attempting to study the sounds, and I just cannot make most of them in a way that Danes can understand at all (e.g. saying place names, even those I've practiced!). To a somewhat lesser extent, I also have trouble parsing them, though I can recognize some stock phrases that I've heard often. Oddly enough, the Swedes I know can't actually communicate with Danes in spoken language either, despite written Danish and Swedish being nearly identical. They will typically switch to English to talk to each other, even though officially they're supposed to be able to communicate (Danes<->Norwegians and Swedes<->Norwegians seem to have more luck).
> Oddly enough, the Swedes I know can't actually communicate with Danes in spoken language either, despite written Danish and Swedish being nearly identical.
It's not that odd. Take threedaymonk's example: English is written almost the same way in New York and in Glasgow, yet an American in Glasgow might have some difficulty understanding people at first. And of course mutual intelligibility between Danish and Swedish will be lower than between New Yorker and Glaswegian.
Written Portuguese and Spanish are also very similar, and while it's possible to keep a "bilingual" conversation with some effort, I (a Portuguese speaker) and my Spanish-speaking friends just use English instead.
Native Swedish and Danish speaker here. I can confirm that Swedish people usually do not understand Danish, whereas Danes have an easier time with Swedish. I think Danish is just plain harder to understand, and I sometimes speculate that my early exposure to it has kind of hypertrophied my general language ability (I used to win awards for English vocabulary knowledge and that sort of thing in high school).
Regarding the phonology, I'm not so sure they're that dissimilar.
As I understand it, English trickled down from the north of the British Isles. The further north you go in Britain, the more likely you are to hear some of the phonemes you would normally hear on the continent.
From my perspective, being a Yorkshireman, I find it easy to parse and understand the accents (and dialects, even though people say english doesn't have any dialects, which it clearly does) from all over the north of Britain that people below the danish line struggle with, such as thick Yorkshire accents, Geordie and Scottish. For me, that's English. All that stuff below the Danish line is wishy washy french.
I think it works the other way round, too, though there's an asymmetry in the fact that, whilst most Scandinavian speakers encounter English regularly, the reverse isn't true.
Nonetheless, even though I've never studied any of the Scandinavian languages, nor ever been there, I find it fairly easy to read written Danish or Norwegian. I went to a Christmas market at the local Norwegian church last weekend, and amused myself by reading all the Norwegian signs and inscriptions. With so many cognate words and an almost one-to-one mapping of word order, I could easily guess the odd word that wasn't immediately obvious.
(None of that either supports or refutes the Scandinavian hypothesis, of course. It's just an observation that there are significant common features and vocabulary.)