Timelines:
Does anyone have any actual info on the Atlas timelines/waiting lists? I haven't seen a single success story/blog post discussing this. Atlas FAQ states 2 weeks to get up and running (tax ID and payment acceptance), but when asked about waiting time, their answer is "We don't have an exact timeline for adding new users, but we'd love to chat more about your use case".
e-Residency (from application to card delivery) should take 2-3 weeks. Not months.
Taxes:
The difference in the level of complexity of US vs Estonian tax rules can't be described in words :). That's why you will need the huge number of tax advisers/lawyers in US, and you'll pay them.
I'm not really sure any person, who's not a resident of US and is not living there, would benefit from having a body with tax residency there, if there aren't other compelling reasons. For example, you would do this if your funding source is coming from there, because funding is better than no funding, no matter the taxes.
Funding/holding:
US has a huge advantage here, clearly!
Transparency:
Estonia is playing for total transparency and an efficiency heaven. If you want to do shady offshore stuff, then this is not for you. #nopanamapapers
TL;DR: Starting and new VC funded startup or a big corporation -> US. Starting a one-man online business (and currently not a tax resident of US) -> Estonia!