Yes exactly! On Kindles to install that app we need a jailbreak (and the procedure will depend on device and firmware version, since we are trying to circumvent Amazon limits), most of the OS stays the same and you can still use the normal "reader" app.
Of course if you are already satisfied by the normal reader all you need to do to gain more freedom is managing your books with Calibre on a computer, it'll take care of converting to kindle format if you put an epub in it, and send it to device, with just one click. My dad does this after I showed him once or twice and he's not techy at all.
Way to go, modern phone screens are not too bad..I think a big reason I enjoy dumb ereaders is the lack of multitasking lol
Your kindle probably has a busted micro usb port (the cables too can stop working for data transfer), no idea if it's too difficult to solder a new one.. I would try cleaning it up really well first