Apple's Bluetooth keyboard support in iOS is actually a bit more flexible than most people give it credit for, even if iOS's external keyboard support is both a little crapalicious (as you point out) and quite the afterthought on the part of Apple.
There's a lot, frankly, that the hardware keyboard can do -- and also a number of excellent reasons that third-party apps don't support more of that kind of input.
Here's the low-down.
iOS supports many keyboard shortcuts, including brightness and dimming, volume adjustment and playback controls (including pause/play, fast forward and fast reverse). If you're using an Apple-branded BT keyboard, these features appear as function keys (namely F1, F2, F7, F8, F9, F10, F11, F12). A single tap on the function key provides instant responses on the paired iOS device.
So that's pretty cool. Not mind-boggling, but cool.
In addition, iOS understands a number (not a lot mind you, but a number) of command-key shortcuts. For example, you can use Command-A, Command-C, Command-V and Command-X to select, copy, paste and cut data from any compliant text view. Command-Z adds in handy undo support as well.
Yes, you have to tap on the text view to get the action all started, but that's right in line with the way iOS normally handles interface interaction. Once editing that text view, all the arrow keys do in fact work, and shift- and option- modifiers allow you to select and move with greater precision.
So there's that.
Unfortunately, iOS was never really meant to offer keyboard control. It subscribes to what is a see it/tap it/do it philosophy. So, no, you cannot move to elements in the interface using standard controls.
However,... there is actually a way to navigate and use the interface from the keyboard.
That's by using Voice Over. You know, the accessibility feature? Believe it or not, it is the most keyboard friendly component of the entire OS. If you want to explore, you can do so by enabling Voice Over in General > Accessibility preferences and then use the arrow keys and shift modifiers to navigate through and interact with on-screen elements.
It's a bit of a pain, so I'm not exactly recommending this as a solution, just letting you know it exists.
As for third-party developers, there's a reason very few of them support keyboard equivalents, and that's because applications cannot read key presses without creating text aware components. In use, that would force the on-screen keyboard to appear if the Bluetooth keyboard is not paired.
Yes, there is a solution around that, but not one that Apple would welcome on the App Store, so the consequence of this is that most devs omit keyboard support, not even taking into account the vanishingly small customer base served by such modifications.
Okay, summing things up:
Keyboards were late to iOS, acting mostly as a courtesy afterthought
A few keyboard powers are available for use, but they are limited
Third-party devs have good reasons not to support keyboard equivalents in their apps
Sorry that Auntie can't really offer the solution you're looking for, but you might want to file a feature enhancement request with Apple rather than trying to track down Santa Steve to make your requests.