[gumby]

I think you have a fundamental bug in your model of the funnel. The wide end of the funnel is the people who hear about the site and arrive there no matter what the means. Those who sign up are the ones advancing to the next stage (your clip art only has one stage, which may reflect your misunderstanding). So putting up an immediate sign up doesn't cut the number of people who come in through the top (as the article says) instead it just makes the first stage of your funnel shorter.

Now the aim of each funnel stages is to discard as many unqualified buyers as possible as soon as possible to avoid wasting your (and their) time. But how soon is "possible"? You have to qualify them (give them some content in your case, and see if they engage) to see if you are a good match. If you push the sign up right up front yes, more people may advance to the next stage, but you have no idea as to the quality of the leads (%false positive and %false negative).

Consider the rediculous extrema: you could assign everyone a user id when they first arrive (basically: a cookie!) which would give you a 100% conversion rate. Or you could disallow any sign up at all: a 0% conversion rate. It's easy to see that these boundary cases are useless.

But how do you know that your new approach isn't equally as useless? All that mathematics is exciting but doesn't address the two core questions: how many of these sign ups became revenue generators and how much revenue was abandoned to people who wouldn't sign up?.

TL;DR: You've analysed how quickly you get out of the driveway without looking at how that relates to getting to your destination (say, if you're even turning in the right direction).