Who paid +1k to own that hardware? I did. This isn't a charity. I do whatever the heck I want to do with my device. Apple can offer an App store, but should not be enforced into using it.
Sure you can, you can use the device as a doorstop. There's nothing stopping you attempting to install whatever you want on it. You could start compiling apps for it manually for example or Jail break it.
But you want Apple to make that easy for you.
iOS is not a multiplatform OS like Android, Linux or Windows. It's designed to work with a small number of specific devices and work extremely well.
The second you start opening up the OS you are bound to get similar levels of discontent as you do with Android Apps e.g. sluggish functionality, broken apps etc which subsequently will lead to the customer thinking that all Apple devices are crap thus irreparably damaging their reputation (like Windows for many years).
And then there's the developers, who will not necessarily gain from this, but completely lose out on revenue altogether due to apps being side loaded from dodgy sources.
You want an example of this ? Just go to Piratebay and type in "Android Apps" and you'll see repositories released every month of Paid Applications.