Found a bug when changing my wifi password and thought I'd post it here for visibility / if others are having the same problem.
When I changed my wifi password, I went to turn on my roku, which turned on fine saying internet not connected, expected. However the remote was working very intermittently, every 1-2+ minutes I could enter input for like 2-3 seconds. During this time power and volume worked the entire time, which have been linked to the TV so I was able to rule out batteries (also changed batteries to be sure).
Reset the roku several times, repaired the remote several times, same issue. Finally I just bit the bullet and spent the 30 minutes it took to enter my password in the intermittent intervals where the remote worked. The SECOND the roku was connected to wireless again, the remote functioned perfectly and has not had issues since.
I assume there is some process either spin retrying to connect on roku, preventing the input from remote, or some process for some reason requires internet to properly receive input from the roku.
Either way, from a long period of testing and turmoil it appears that the issue is mostly definitely linked to the loss of the internet connection. (Specifically through a changed password, though I don't want to go through testing this in other methods).
There are a couple of things that could be going on. One could be that the remote is bad, and it went bad coinciding with your WiFi password change, or during the time since.
Assuming the remote is not bad, we simply need to force it to connect.
First, I'm assuming you removed the batteries from the remote during the time it was not inn use. If you left the batteries in, it's possible the contacts are corrupted. Ensure there are new batteries in the remote, then push the pairing button that's behind the battery cover. Does a light flash? If not, the batteries are bad, the batteries are not connecting (battery contacts), or the remote is bad.
If it flashes, the remote is getting power. Remove the batteries.
Next, do another factory reset of your Roku Stick (3600). Hold down the reset button for at least 30 seconds, while the Roku is plugged in to power. It doesn't need to be connected via HDMI, just power when you hold the reset button. Oh, and make sure you hold it continuously for a full 30 seconds.
Next, unplug the Roku from power, connect it to the TV via HDMI, then apply power.
Next, put the batteries in the remote. Be sure you have them properly aligned. I don't recall if they are both in the same direction or different directions on the 3600 remote. Just ensure you have them properly aligned.
You should be going through the same setup as when the device was new. Simply continue as if this is a new device setup.
If the remote simply isn't working, you'll need a new remote. The Roku Voice remote will work. It's $20 and is available for order online from Roku, but you can also find it at Walmart, Best Buy, and other places that sell Roku devices. The $30 Enhanced Voice Remote should work, too.
Now, before you spend $20 or $30 for a new remote, consider how much it would cost to replace the Stick entirely. Currently, the Streaming Stick+ is $50, but you may find one on sale somewhere (though I'm not finding a sale).
Consider that your device was discontinued in 2017, and that the current Sticks are noticeably faster, you may want to consider an upgrade. If not, and you're otherwise satisfied with the 3600 Stick, a new remote would solve the problems, if the current remote won't work.
DBDukes https://www.dbdukes.com/Roku Ultra (4660) Roku Streambar (9102) Roku Streaming Stick+ (3810)