The info button does not appear in the device setting under 'My Devices' when the BLE(peripheral) connected programatically. Another issue is when the BLE get disconnected programatically the device get removed from the list of connected bluetooth devices in setting.
Not actually the answer, but hope this information will be useful. One of the most frequent reasons to use (or at least to find) "Info" button for BLE devices is to set or restore "share system notifications" flag that allows system notifications (calls, messages etc) to be passed to Bluetooth devices. For example, a lot of people who use different smart devices (fitness trackers, smart watches) are having issues with such notifications after upgrading to IOS 13 (there are many topics all over the Web - both vendor-specific and generic). And there are some screenshots where "LE" devices (according to label) actually have (i) button next to "Connected"/"Not connected" (e.g. https://www.samsung.com/us/support/troubleshooting/TSG01202131/)
