Yes, in my experiments removing the free card basically killed the conversion funnel, so your marketing costs per paid user skyrocket. Not only that, but the people who did convert paid for a single postcard instead of bulk credits, obviously, since they wanted to try it out first before spending big bucks. Bulk credits are essential for the business to even have a chance since you need to get your paid users to send (or at least pay for) at least 10 or so postcards for you to break even as I mentioned above.

edit: The 10 postcard number here is more of a guess if you assume that by removing the free cards you've increased your overall marketing costs per paid user to offset that cost anyway. That said this model can work better if you assume marketing costs are zero, such as if you get to the top of the app store. In the real world they aren't but in theory Postagram (who is at the top) could turn off free cards and start eeking out a legitimate (read: not based upon credit expiration) but small profit.

What if you made the app cost $0.99 and that included the first postcard?

You get far, far fewer purchases, but each purchase is a far, far more qualified customer, plus, you can almost break-even on the one-and-dones, and you may make enough money on the breakage to make it tiny profitable.

I get that tiny profitable isn't a VC's goal, but for a side project that's dying anyway, this might make the runway infinite, which you could use to rattle around and maybe find a winning model, or just let it run and not cost you much.

Problem is the app gets no downloads organically. It costs me 0.50 per install for the free version via Facebook ads, and this is pretty good.

What if you had a great product video/detail shots/testimonials about the quality of the print?

I'm a photographer, and so I'm picky about print quality, but I would be willing to roll the dice on some gift-type prints, and I could easily be won over by some nice shots of a quality print.