Rooter? Or router with the out pronounced owt?

I am not that much of a geek but I get the song. It bothers me though that the singer says "rooter" not "router" with the out pronounced out as in the great outdoors. "Owt" not "oot"

I worry that I may be mispronouncing it... mostly in my brain. It is not like I sit around and talk about routers and servers... but I am pretty sure when I have talked to the Real Living Help desk we were all saying routers not rooters (rhyming with Hooters, a dining establishment....) Router pronounced "rooter" also makes me think of Roto Rooter.

When I took the ePRO class I pronounced it routers not rooters... was there audio in the ePRO course or was it just reading the screen? When I've read PC's for Dummies, Internet for Dummies, etc. I have always read it routers not rooters.

When it comes to the word Route I am kind of torn. Route 66, the famous song tells me to say the word "Root 66" and so I do. Yet if I was going to follow that famous route cross country to me it would still be a route not a root. Trees have roots. A carrot is a root. You follow a route. If I follow a route I get there. A path, a road,

To me data packets take routes not roots.

Router from Wikipedia:

"A router is a device that extracts the destination of a packet it receives, selects the best path to that destination, and forwards data packets to the next device along this path.[1] They connect networks together; a LAN to a WAN for example, to access the Internet."

And more concisely and yet still from Wikipedia:

"A more precise definition of a router is a computer networking device that interconnects separate logical subnets."

I was sure "route" and "root" would be questions in this American Accent Quiz but they are not. I wonder if there were people troubled by the Route 66 (pronounced root) song back in 1949?

Oh here's the router - rooter song...