it's not a paper form, so it's not possible to "not have it". From reading the news article the provided link points to (you're welcome to read it yourself), there will be "help points" at airports upon arrival, they're there for "exceptional circumstances only". Reading into that, I don't think forgetting is an 'exceptional circumstance' and, just my thoughts, one 'could' be refused entry. Like everything else with Thailand, it may well be up to the Immigration Officer you're in front of.
a link was provided to a video and explanation of the new card. Did you click and read? Your answer is there. You will need to complete the online form, for as you said you'll almost certainly get to passport control/immigration just after 12am on May 1. You can't complete the form before it goes live, so yes, on/after April 28. Don't forget it though. While the news article says that there will be "help points" at airports on arrival, for those who haven't completed the form, it's for "exceptional circumstances only", so if someone has just forgotten to do it they may be refused entry.
Non-OA Visa application fee is about double the price, requires a Police Clearance, Health Insurance and maybe one or two other things that the Non-O doesn't require. It also has to be applied for in one's own country of residence, where the Non-O, you can apply for within Thailand, or any overseas Embassy or Consulate.
will correct me if I'm wrong (or delete my comment), but the Non-O Visa is still available at Embassies/Consulates overseas. I think it was the Non-O 12 month Multiple Entry Visas that are no longer available.