I'm an App development studio owner only and have worked across native and cross platforms, and also on so called magical tools, so let me throw my 2 cents -
When you have a decent product on web already, then go for native app only as the quality offering to your customers. I'm yet to see a successful app made on such platforms and in the easiest way and cost they claim they offer. The reality is, you end up with more time, cost, and pain with these alternate routes if you are really launching a commercial grade product to compete with other apps in the market. And then, a user once uninstalls an app will hardly come back.
Scaling up these cross platform apps is a big problem as well, so as integrating with essential third party SDKs for example flurry. We recently tried integrating flurry events capturing in a phonegap app and it was a nightmare, the same job could have been done in 10 hours on iOS native or Android app.
So bottom line, if you want an app just for the sake of having it with no big goals then you may think of cross platform, but if app is a part of your growth strategy and is a serious effort then go for native, no matter if for one platform at a time, iOS or Android.
$ 10 K is an okay amount for an native App development based on my experience on what we charge, when server side is ready and we only have to work on screens, API integration, performance, design etc. I can give you many examples of nice apps created in this price range.
Using parse or urban airship are good options also if used with native apps, though if you have an existing product and must have relevant infrastructure too, then go with your everything custom for long run benefits. Parse becomes costly and cumbersome when number of users increase, and migration will also become difficult, so think wise way.