What to do in the first ten minutes
Leave Mixpanel exactly where it is. Add the Tiny Funnel script tag next to the SDK, then open the recorder and click through your own signup the way a customer would. The landing page, the pricing page, the signup form, the welcome screen. That click-through is the funnel. There is no event to name, no release to ship and no data to wait for.
If the last step of your funnel finishes on a server rather than in a browser, a subscription webhook or an invoice paid on a schedule, post it as a server event with your own user id and it lands on the visitor who caused it.
Why the two tools will disagree, and why that is fine
A Mixpanel funnel is a sequence of events, and an event exists because somebody decided in advance to record it. That is not a flaw, it is what makes the deep analysis possible, and it is also the cost. The funnel can only be as good as the taxonomy underneath it, and the taxonomy is a meeting, a naming convention and an engineering ticket before it is a number.
A Tiny Funnel step is reached or not reached. It can be a URL, a click, a page state or a server event, and it was recorded by clicking through the site rather than by instrumenting it. Nothing had to be decided in advance, which is why the first answer arrives in minutes, and also why there is no retention curve behind it. Two tools, two questions, and the honest position is that a product team at scale wants both answered.
The short version
Keep Mixpanel for how people use the product. Use this for who arrives, how far they get and who buys. Setup is one script tag and a click-through, and the two tools disagreeing is information, not a bug.