It's a trade-off. You can make a longer, less-pithy sentence to get begin/begun to agree with both modal verbs, or you can get a terser, more-pithy sentence if you're willing to live with one violation.
The short version:
Revolutions can, and often have, begun with reading.
puts a lot of stress on the word have. Reading it aloud, the word gets the highest intonation of any word in the sentence, and it gets drawn out and followed by a pause. The present perfect tense says that the deed is done and is still relevant. But we still haven't gotten to the deed! Finally, "begun with reading" says what what this big deed is, now that you're expecting something important.
This puts huge emphasis on the fact that revolutions beginning with reading aren't just a speculation, they have actually happened. Of course, this sentence needs to follow or be followed by facts about some actual revolutions that began with reading, or the reader will feel gypped.
The long version:
Revolutions can begin, and often have begun, with reading.
seems less emphatic to me. Repeating a word sometimes gives it more emphasis, but here it just seems to blur the point. A pause now separates "begun" from "with reading". The main point of both sentences is about reading, but this version distracts from reading with two forms of the word begin. The short version is simpler and stronger.
It often happens that a word later in a sentence needs to be in two different forms to agree with two words that came earlier in the sentence. And it often happens that making the sentence longer to do that makes it clumsier and less memorable. In this battle between perfect grammar and rhetorical force, proximity agreement awards the victory to rhetorical force. Just make the later word agree with the more recent of the earlier words—especially if the earlier word comes immediately before.
Native speakers don't even notice that this is happening. One tends to notice it only while writing, when one is examining every word and nuance carefully. It's like the way a stage magician can hide a card or coin right in front of your eyes, but you don't notice that anything is amiss because the magician has distracted you with something else. In this case, the agreement between have and begun sufficiently distracts the reader from the disagreement between can and begun that the reader doesn't notice.
And yes, it's perfectly fine in writing, no matter what that article says. Good writers do this all the time, because good writers don't sacrifice rhetorical force to grammatical nitpickery.