The one thing they have correct is that aggressive subject lines are generally better at getting click-throughs than milder ones. "You'd be crazy to miss out on this" works better than "Check out our new product."
I used to work for a flash sales website, and whenever I told people about it, they'd uniformly say "Oh yeah, I've signed up for them! Way too many emails, can you make it stop?"
I'm sure the folks at Weebly have tested this (or maybe this poor unfortunate soul is part of some silly week-long A/B test to try and improve engagement.) But my personal experience says that Weebly needs to get smarter about how to engage.
More effective: Send someone a couple of welcome emails. Wait 4 weeks. Then send them a message. BAM, lots of engagement, because when you receive the email you say "Huh, this must be pretty exciting, I haven't heard from them in a while!"
Have you ever signed up for Everlane? Go do it and pay attention to the way they do onboarding and email marketing. They don't spam me, but when they do, I'm like "NEATO LET ME SEE YOUR NEW CASHMERE T-SHIRTS" because it seems like a rare opportunity to see a budding startup release a new product. It's all marketing but done in a way that doesn't suck horribly.