Big Foreigner songs you left out: Hot Blooded, Urgent, Head Game. Honestly, I always thought of them as the definitive “generic late ’70s radio rock” band, at least kind of in a bad way. They had songwriting chops, but I always thought that they sounded like a bunch of session men.

Regarding the comparison you’re making, Bad Company’s best known songs are from 1973 & ‘75 and don’t have the late ’70s sheen. I think REO Speewagon sound more like Foreigner than Bad Company. The band that I’d think of Bad Company sounding like to me is Grand Funk Railroad, probably because I always get Somekind of Wonderful confused with Can’t Get Enough of Your Love. Grand Funk was an early-to-mid 70s band too.
Another hit I’d expect people to know by Speedwagon is “Time For Me to Fly.” Their album Hi Infidelity went diamond.

Bad Company songs I’d expect people to know:

“Can’t Get Enough”
“Feel Like Makin’ Love”
“Shooting Star”
“Bad Company”
“Rock ‘n’ Roll Fantasy” (okay, this one is late 70s and sounds like it)