So I use the app from Cleanbrowsing and gave the accessibility permission to it and saw that it can view my screen, action and inputs, perform actions and control the display.
Is this app even safe to use?
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It always depends on how trusted the app is.
As for why the accessibility service (which this permission is about) has such powerful control, it is because the original purpose is to help people control their device more easily. Taken from the official Android documentation on Accessibility service,
An accessibility service is an application that provides user interface enhancements to assist users with disabilities, or who may temporarily be unable to fully interact with a device.
Example of accessibility services:
However, developers also found unusual ways of using Android accessibility services, as explained by Maksym Dybarskyi:
[...] there are several ways of unusual usage of accessibility services. I wrote ‘unusual’ because all these scenarios are different from the original purpose of accessibility services.
Automation apps like Tasker, MacroDroid, and Automate have their own "accessibility services" to allow more features that are otherwise restricted by the system (e.g. intercept action on buttons, take screenshots, etc.)
As for CleanBrowser, the documentation claimed,
Advanced functionality includes removing the users ability to uninstall the app. [...] it makes use of the Accessibility API to prevent the user from uninstalling the [app],