You should maximize the "wow" moment your product delivers. Since you say it is heavy on "server integration", I guess you would need to focus on monitoring & server capacity and of course, features that directly address your users' problems instead of fringe, nice-to-have add-ons. Everything else (analytics, payments, non-core features) can be left out or done in a way that minimizes effort (Stripe + Google Analytics should take up very little of your time)

Your users are not all the people who visit your site because of the "launch" but those for whom you can solve "the problem" well. I will definitely go back to a product that solves my problem, even if it has a lot of bugs. As proof, I've used buggy UML tools, code editors, video/audio editing software for years because they're very good in at least one aspect I value.

A lower-risk approach: Suppose you list out the benefits of a "launch" (example: growth from 0 to 10K users, investor interest, attract/keep talent) and then separate out the components of a launch (example: Techcrunch release, beta mailing list, investor reach-out). You may be able to de-risk by going after individual components and only building whatever is necessary for each individual component. For example you can attack the mailing list first without worrying about scale to handle traffic that a press release would bring.