Montrose
The Larry Mullen Experience.
Larry Mullen says that was the name of the band he was putting together would be … until Paul Hewson walked into the room. Today they are U2.
Then Emerson, Lake and Palmer 2/3 counts. Although Blackmore’s Night is a much more interesting name than the law firm name of ELP.
Also Mick Fleetwood’s Zoo
Robin Trower didn’t sing on his albums, that was James Dewar.
Argent, maybe? Founder Rod Argent did sing some lead vocals, but it looks to me like if the band had a lead singer, it was Russ Ballard.
Yngwie Malmsteen ( OK, Yngvie Malmsteen’s Rising Force )
Brinsley Schwarz - notable for the first recording of (What’s So Funny 'Bout) Peace Love and Understanding.
Eddie And The Hot Rods (but only if you accept a mannequin as a band member).
j
J. Geils Band
Steely Dan was a honorary member of that group. The ladies loved him. 
The OP question is difficult. Bands create unique names like Genesis, Rolling Stones,Beatles or they name it after the frontman vocalist. It’s rare to name a band after sidemen.
I think Doctor Hook was named after the guy with the eye patch. Eyepatch/pirate, It’s their sense of humor. Dennis Locorriere is their vocalist.
Is Ted Nugent the name of a band, other than the guitarist?
Well, there are quite a few examples in this thread. Now I’m wondering if it’s actually more common than bands named after the lead singer (if you exclude band names of the form “Somebody and The Somethings”).
Also, I’m not sure about your use of the term “sidemen” here. According to some definitions, a “sideman” is a hired hand, not a regular member of the band.
I assume you’re using the word “sideman” as a contrast to the “frontman,” but even so, it seems somewhat unnecessarily dismissive. There are examples of bands in which some other member is at least as important, if not more so, than the frontman/lead singer, including arguably Pete Townshend of The Who, Rick Nielsen of Cheap Trick, and Tom Scholz of Boston.
Sidemen play important roles in bands. I have a lot of respect for them.
Nick Mason’s Saucerful of Secrets – Gary Kemp handles the vocals.
Along the same lines, Valerie was described as being a record by Mark Ronson ft. Amy Winehouse - I think Ronson did/does that quite a bit.
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Bruford
Bill on drums, Jeff Berlin sings, and plays a wicked bass.
The Alan Parsons Project
Alan Parsons was the producer.
That was always my understanding but according to wiki, his roles included:production, engineering, programming, composition, vocals, keyboards, guitars (1975–1990)
Yeah, that still tracks. A lot of famous producers also did a lot of otyer stuff. George Martin wrote and played a number of parts on Beatles songs.
Alan Parsons is a producer primarily. He has worked with a LOT of acts.
Is Santana or Van Halen the name of a band, other than the guitarist? If only the last name is used it elevates it to band name?