I agree with ஒழுக்கின்மை's list 100%. I didn't want to post a long wall of text and that is why I posted only the links... but here goes:
I haven't released anything on the mobile world (yet), but I have read postmortems and checked a lot of posts from various developers as well as conferences (namely Casual Connect) for the past few years. Here are the there ways any game has succeeded so far:
1- Lucky enough to get a publisher.
In this case, you are either a lucky company/person where a publisher notices your game. The publisher initiates the contact and asks you if you need one. You agree on the terms and that is that. The publisher takes care of everything there is about the game marketing, you just have to give them a percentage or whatever the deal it. In some cases, they'll request that you'd implement some kind of API if they have that kind of presence in the App Store.
2- Lucky enough to land a publisher.
This is the more common one when you don't want to do the dirty work yourself. You get a list of publishers on the App Store and start mailing everyone of them about your game. However, you'd need a good following or a VERY interesting game with "social" potential to land one. So if your game doesn't generate some kind of income, there is doubt a publisher would be interested (again what I read). So in your case, if there are no IAP (In App Purchases) and you are releasing it for free, don't bother contacting King to publish it for you -- for example.
3- You become the publisher.
This is where you are now. You gather the fans, you market the game, you do everything basically. It is the hardest form and the one that requires A LOT of luck (or at least some connections).
This article details how you can get the press to notice you. Despite that, there is no guarantee this works. In the end you can contact everyone on earth and it doesn't work out (I for one haven't heard of the game in the article until I read the article... though it isn't like I know every game there is in the world... but just so you know how it may end). So you'll need prior to your release to have everything set up (presence on Twitter, Facebook, Google+, IndieDB mobile version, GameJolt, your own website with a forum of your own for example, etc...)
Once you've done that, you start generating a buzz through trailers, competitions, offers, beta tests, etc...
This should provide a ground for testing the waters so to speak, if you have a good number of following you can hit something like Epocu and see how long it takes you to hit your target. The time it took you to hit the target and the size of the target should give you an idea of where you stand in terms of popularity... there you decide either you want to hit the market right away or wait for more buzz.
There is also number 4... but I doubt it applies to this situation... but it is the Tim Schafer effect. As ஒழுக்கின்மை mentioned, your popularity and track record. Tim Schafer created a Kickstarter campaign, had basically nothing to show people and all he said was something along the lines of "we are going back to the roots of Monkey Island, come back me up". He got 3.3 millions while asking for 400K. Why? Because people have been a fan of his work for a long time and have been wanting to get a new game like what he used to create.
The good news with this is, if you decided to go "the famous person effect" route... you can. You just need some connections with someone known to players. So let's say (for example) if you know the designer of Angry Birds and he showed up in a trailer or posted on your blog/Twitter, this game is cool count me a player, now every single Angry Birds player will be jumping hoops to get to play your game (in a sense and not literally).
Read Derek's Yu talk about the remake of Spelunky... if I recall correctly Jonathon Blow (creator of Braid) helped them out a bit with an extra push (albeit the game was great as is).
Yet, all that is nice and dandy but without quality of the trailer / website / promotional materials as ஒழுக்கின்மை said... it won't work as good as it would be expected. Yes, it might work (Snood is an example as he mentioned before) but you don't want to take a gamble... you want to do everything perfect so your chances would align better.
I understand that you read that 67% of the downloads came from direct search in the App Store... does it tell you that they were searching for the exact same name as your game? As far as I know the word "Tappy" is a pretty common name (one of the first clones for Flappy Bird was Tappy Bird).
After all said and done, whatever we say doesn't matter. I believe some of us are still trying to understand this whole thing as much as you do. We aren't here to bash you, nor are we here to say this is wrong or this right (at least I know I am not trying to do that), we are just trying to give you a different perspective and a different way to view things. We just hope it may help. Since if you succeed then we have a hope to succeed too, I guess
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PS: Sorry for the long boring post ^_^
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