https://www.nature.com/articles/d41586-022-00215-2
"Omicron forms a stronger grip on ACE2 than do previously seen variants3. It is also better at evading the virus-blocking ‘neutralizing’ antibodies4 produced by people who have been vaccinated, or who have been infected with earlier variants [..] At some point [..] something happened to help Omicron explode, maybe because the progress of other variants — such as Delta — was gradually impeded by the immunity built up from vaccination and previous infection, whereas Omicron was able to evade this barrier."
Evading vaccination could very well be why it was so fast to spread in mostly-vaccinated populations in Europe and North America, which happened "faster than any previous versions" as the (very interesting!) article you linked says at the very beginning, but the variant was more fit anyway.
First, Alpha was from Dec 2020-Jan 2021 in the UK and was already substantially more infectious than the Wuhan wild type or D614G variant, though it did not escape immunity.
Second, as far as we know neither Delta nor Omicron (nor all the other Greek letters that were outcompeted by them) developed in vaccinated people, and both of them are anyway incredibly more infectious even in unvaccinated people.
I don't know about BA.4/5 but, at least for the three main lineages of concern that drove waves after mass vaccination, selective pressure was not a cause of either the birth or the spreading of the variant.