First a correction regarding the underlying facts:
- It is misleading to say that standard German does not have a progressive aspect. As pointed out in a passage you quoted, standard German has an extremely short and efficient method to express the progressive aspect: adding the adverb gerade, which roughly translates to right now. It's just not obligatory. And most German dialects have additional ways of marking progressive aspect that are structurally more similar to how it is done in English. Moreover, for intransitive verbs the Dutch style of marking the continuous with daran sein, etwas zu tun (literal translation: to be at it to do something) is well established in colloquial German of most regions by now. (All not obligatory.)
- Another feature of colloquial standard German that got some attention (and its name) rather recently is the absentive aspect. Structurally this is about as similar to the English progressive aspect as possible, the only difference being that it uses the infinitive rather than the present participle, just like the Dutch progressive does. (The commonality is that in all three languages it's the verb form that functions ordinarily as a noun.) The real difference is (for now) in what it expresses. The absentive typically expresses that someone is in a different location right now in order to do something there that can only be done there. E.g., if we are in my kitchen and you ask me where my wife is, I might answer "Sie ist kochen". This doesn't just mean that she is cooking right now; it implies that she regularly goes to some other place in order to cook there, and that's where she is right now. Dutch also has this construction with the same meaning.
- It is misleading to say that English has the progressive aspect in common with most other Germanic languages and many other languages. In English it is (essentially) obligatory to mark this aspect, i.e. in most cases when it can be marked it must be marked. Whereas ways of marking progressive aspect abound in all sorts of languages and in fact many have more than one, this obligatory nature is actually quite rare. In particular, English is the only Germanic language which has this phenomenon.
So English is about as odd in this respect as German.
Proto-Germanic did not have a progressive aspect. It also did not have a future tense or a composite tense like the present perfect. This all came up later, mostly in parallel in many European languages (including non-Germanic ones), probably under the influence of Latin and each other.
Latin (and French)
Latin has had a lot of influence on the evolution of Germanic languages, not just as the ancestor of French, but also directly as the scholarly lingua franca for many centuries. When scholars wrote grammars of languages that were previously not standardised, they were generally influenced by Latin grammar. Latin does not have a progressive aspect in the same strong sense that English does. It has two forms of the past tense (perfect and imperfect), one of which carries the progressive aspect, and it permits optional marking of the progressive in any tense by using present participles as in English. French has être en train de faire quelquechose (be in the course to do something), but this is quite clumsy and not obligatory.
I guess that when German scholars tried to establish a standard variety of the language for all German speakers (an attempt that was successful except for the Low Countries opting out), a literal translation of the Latin progressive using the present participle sounded too wrong to be seriously considered. Gerade is quite satisfactory for expressing this aspect anyway, and the various other dialectal ways of doing it were quite different from each other and not really a good basis for the standard. Side note: Dutch covers a much smaller region for which iets aan het doen zijn (be at the do[ing] something) was a common alternative to nu (now), so there was no real obstacle to it becoming part of standard Dutch.
How progressive aspects and similar features come up
Grammatical features evolve naturally and gradually, often out of idioms. Maybe some people once started saying something like "She is in the count's kitchen in order to cook" rather often until it became an idiom. Idioms initially start as expressions that can be understood by straightforward analysis, but then they are used so extensively that it becomes possible to (a) shorten them in such a way that an analysis no longer makes sense, and (b) extend them to situations in which the original analysis doesn't make sense. (A good example for (a) is "I could care less", which is the result of removing the negation from an idiom. Any figurative use is an example of (b).)
This is how the above example sentence may have been abbreviated to "She is in order to cook", then "She is to cook" and finally "She is cooking". Here you need to understand that even English once had infinitives that were clearly distinct from finite verb forms and more similar to the present participle, so that it was quite natural to switch from cook to cooking at some point, as an analysis of the short final form is closer to making sense with cooking than with the infinitive cook.
Note that I am not claiming that this is how it happened. To the extent that the history of this construction is known, it is actually a bit more complicated. For anyone who wants the real details, see pages 6ff of this paper. This is just an illustration of how it could have happened.
How aspect marking can become obligatory quickly
Speakers of all languages like to mark progressive aspect because it tends to make statements livelier and stronger. So there is a natural tendency to come up with progressive aspect markers and then use them more and more over time. However, if (or so long as) they are as clumsy as aan het doen zijn, they have little chance of becoming obligatory.
Among the Germanic languages, English is in many respects the one with the fastest development. (A fast evolution is often seen in languages that are the synthesis of several different languages. English started as the German dialect spoken by the Angles and Saxons who invaded England, then it was adopted by speakers of Celtic languages and influenced further and rather strongly by speakers of Scandinavian and French.) This is one reason why it's natural that its progressive marker was simplified faster. Another is that a lot of English speakers used to be bilingual with a Celtic language, and as far as I know these generally have obligatory progressive marking as well. So they will have pushed towards overuse of the progressive and thus indirectly to simplification.
Once it was simplified to its current state (or nearly), likely there was no real obstacle left to making it obligatory, and at that point the Celtic speakers probably treated it as such in any case.
How aspect marking can stay non-obligatory for a long time
Dutch-style progressive marking is still very clumsy in German, and extremely so for transitive verbs. Clumsy constructions generally don't get the kind of overuse that permits them to become obligatory, and only working for one class of verbs doesn't help, either. Add to this the fact that Latin doesn't have progressives, so scholars were prejudiced against them, and the generally conservative nature of standard German - and you have an environment that is quite hostile to obligatory progressive marking. (Standard German has always - to some extent - felt to speakers like a written language that one can also speak, or in other words as a language of culture that lives alongside the dialects. In yet other words, the way Germans treat their standard language has some similarities with how one treats a foreign language. This is probably why standard German is slower to follow the trends of colloquial German than English and recently Dutch are.)