To me that indicates they have been able to turn off alphabay for a long time, considering how easily and well timed they did it. That also means they have had tons of time to build the case. Of course you can argue that the simplest explanation is the best one but considering law enforcement literally operated the biggest DNM for months, completely under the radar I'm not sure why "they found an email he used for a few weeks 4 years ago" would be more simple.
You can read what DeSnake, another admin of the website had to say about the takedown. He's extremely security conscious (he hasn't been caught yet afaik which is another can of worms) and he's adamant that it was not a simple bust. Actually, the whole thing was kind of a mess, with some mods getting arrested (even without making obvious mistakes like Alex did ). You can read up on the confusion here: https://www.darkowl.com/blog-content/alphabay-marketplace-re...
If I had to guess, some mod/admin informed on him (maybe even snake!) hence why they had access to an early email. But who knows? Now in cases like the silk road I'd agree that it was simply trash OPSEC but the Alphabay/Hansa takedown was so sophisticated that anything is possible