Smashing Pumpkins
Information Society
Emily Dolan Davies played drums in the Darkness for a bit.
Ash - Charlotte Hatherley on guitar
Belle And Sebastian - Sarah Martin, multi-instrumentalist. She sometimes sings lead, but Stuart Murdoch is the main singer. And also Isobell Campbell, former member on keyboards. She also sang some leads.
When I first became aware of Rock in the seventies, this was still exceptional, but I smell a trend that it became much more prominent starting with the Punk explosion, during the eighties and then the second (Nirvana) Punk explosion.
And for an earlier example, what about Christine McVie? Sure, she often sang lead, but she never was THE lead singer of Fleetwood Mac in any incarnation, but mainly the keyboard player.
The Thompson Twins.
If it’s okay to include someone who was a lead singer but not the lead singer, you got your Kim Gordon with Sonic Youth. She took her share of lead vocals, but so did Thurston Moore, and Lee Ranaldo occasionally got into the mix as well. And Gordon was pulling her weight instrumentally with a bass and/or guitar.
Considering all the bands over the years fronted by a female lead singer with men playing all the instruments, is there one where the opposite is true – women play all the instruments but the lead singer is male?
(And no, the “Addicted to Love” video doesn’t count. 
)
Stay Human, which is the house band for The Late Show with Stephen Colbert, currently has two female musicians in their lineup: Endea Owens (bass) and Nêgah Santos (percussion). While I think that both of them do provide vocals, the bandleader (Jon Batiste) is their lead vocalist.
The Cramps were my very first thought. Female lead guitarist (and female bassist, depending on the incarnation) with male lead singer.
The Applejacks were popular in the UK for about 18 months (1964-65). The band featured Megan Davies as a non-singing bass player.
Helen O’Hara was exclusively a violinist with Dexys Midnight Runners from 1982-87.
I just remembered Arcade Fire. Male lead singer and multiple female performers on instruments. Régine Chassagne is actually married to the lead singer Win Butler.
Tony Orlando and Dawn.
Robert Palmer’s videos gave the impression that his backup band was female. I don’t know who he actually performed with onstage.
Oooh, that reminded me: Lisa Germano was the violinist for John Mellencamp’s band on several albums and tours in the '80s and '90s. It looks like she was primarily (if not solely) a violinist on four of Mellencamp’s albums, and then was a background vocalist on a fifth.
She’s featured in the video for the song “Paper in Fire”:
I’m gonna go with Lil Hardin Armstrong, who played piano in her husband’s band. You may have heard of him, he played the trumpet.
Kirstin Maldonado, the only female in Pentatonix, sings lead occasionally, but they’re really an ensemble group.
The Dead South had Eliza Mary Doyle as their banjo player for several years.
I don’t follow him but I remember being surprised that a woman drums on this Lenny Kravitz hit:
I’m no Jack White fan, but this is just mean!
Similarly, Boris. Wata has done vocals, but Atsuo is really the lead vocalist. I think that’s a good thing, because it lets her completely murder with her guitar parts.
Not sure if I’m being whooshed here but the “band” in the video were all actresses and models: Julie Pankhurst, Patty Kelly, Mak Gilchrist, Julia Bolino, and Kathy Davies.
The musicians that recorded the song were all men: Andy Taylor, Eddie Martinez, Wally Badarou, and Tony Thompson.
Here’s Palmer performing the song in a concert that same year. It appears two members of the band are women (plus a female backup singer) and four are men.