I really do understand the desires people have for iOS to be a more open platform, but I'm just gonna say very clearly: I do not want third party apps being able to do what OP's app does. My iPhone is the one computing platform I have where I get the assurance that no third party app can be spying on anything else I do on the device.
Two things can be true, you can choose to install software from a curated store, policed by an entity you trust to do that. I can install whatever trash I want from the internet and risk my own security doing so. These two things aren't in conflict and could be enabled with a change in policy from Apple.
Worried about grandma installing shady apps? Enable parental controls on her phone.
iOS generally lets you reject any permission an app asks for. This would certainly be "risky" enough that iOS would require explicit user permission, and you would be able to say no.
On top of that, the app is completely optional: if you aren't comfortable giving it those permissions, don't install it?
Locking down your phone to block anything you don't want is doable at your own level. Opening up the ecosystem for those who care about better third party apps can only done by Apple.
Those two desires should both be fulfilled.
The Accessibility permission is not granted automatically to apps on the Mac. You have to specifically allow it for an app. So you retain control and assurance even without Apple lockdown.
Yea, Accessibility features are kind of OS super-powers and you really, REALLY need to thoroughly vet apps that you grant those powers to. These apps need to be actually using Accessibility to provide assistive technology for users with disabilities. I'm usually uneasy about Apple anointing itself the gatekeeper for this, but someone has to do it.
Lots of shady and well-known developers (like Dropbox) are notorious for trying to weasel their way into getting Accessibility permissions, so they can do god knows what with them to your system.