Why did English end up with poorer correspondence than other European languages that saw a lot of printing? Was it simply that English included words from a wider variety of languages, and so was bound to be more varied?
Edit: Or is it that English changed more post printing press than other languages? If so, why?
You are actually giving part of the answer yourself (correctly). While u/athozintra above correctly points out that spelling standardization happens only in most countries in the long 19th century, the actual content of which was not necessarily government-driven, the Oxford English Dictionary became the de facto standard for English, just as the Duden in Germany was (in this case by law) granted the right to be "the" defining authority on all German spelling.
However, English is a special case as its vocabulary stem from several European continental languages at once, mixing into the Anglo-Saxon "base" English. The impact therefore of Latin and Norman French words mixed into Anglo-Saxon, Danish and Viking loan words, Celtic, etc. made the harmonization of spelling-to-word vastly more difficult if not impossible. To keep the history of a word it has to retain a certain etymological spelling, which explains what you call "poor spelling-sound correspondence". Precisely because if its joint roots in both Romanic (Norman French) and Germanic-language vocabulary, there cannot be a standardized spelling-sound rule in English.
When I give talks on this matter I usually use the etymology for boulevard as a spectacular example. Notice the two words, "bulwark", and "boulevard".
In old Germanic languages (eg Saxon, Alemannic, etc) , bulwark derives from a German form meaning "Bollwerk", which is itself derived from "Bohlenwerk"-- which roughly translates as "defensive structure made of thick wood (Bohlen)".
Anglo-Saxons used the word bulwark (Bollwerk in German) for any defensive structure, such as a thickly palisaded city wall. So did the Germans in the middle ages.
When French (Frank) traders in the early middle ages visited to trade with German medieval towns, in German cities small shops and craftsmen stalls were placed along the inside of a city wall ("along the bulwark"). In French, therefore, the name for a shopping street was a lean word from the German "bollwerk", which the Franks pronounced as "boulevard". That's how the french word boulevard came to mean an important "shopping street", and was imported into Anglo-Saxon through the Norman invasion after 1066.
Which is why modern English now has both bulwark and boulevard, two similar sounding words with the same origin, meaning two very different things.
If English were to standardize the spelling to "bullwork" or some such, you would not only lose one word, but also some lovely history behind it.