I have a Roku 3 for maybe four years. Have had no problems. Recently did not use the unit for about two months. When trying to use it all I get is the Roku letters dancing around on a blue screen. Remote is showing paired. Has not control. Have disconnected unit several times. Switched TVs. Have reset three times. Still get dancing letters and that is all.
Sounds like it's hung up in the bootup loop. Keep in mind that the Roku 3 dates to 2013, with the last built in 2016. While it's still a heckuva device -- Roku's best ever, in my opinion, they do get old. Mine still works, but it's not totally unexpected for a device that age to be reaching end of life.
If you have another such device -- a Roku 3 model 4200, Roku 3 model 4230, or Roku 2 model 4210 -- try swapping power supplies. If that's not possible, a hard factory reset might be in order, but I'd do that as a last resort. To do that, while plugged in, depress the Reset button for no less than 30 seconds. Don't mind what's on the screen, no less than 30 seconds.
Failing that, it may be time to retire the device. Or, it's retired itself.