GFR hadly ever toured Europe, they were busy enough in the US. By the time they toured Europe, they were past their commercial (if not artistic - I like the later GFR stuff better) prime. There is a similarity in Grand Funk's and Black Sabbath early formative work - that bludgeoning, lava-style approach (Cactus had that too), but both bands left that behind and stretched their wings by their third albums. Grand Funk showed their Detroit/Motown roots more and more as they progressed as songwriters, they became really souldful by the time Craig Frost had become a permanent member and I always thought that made them stand out, but it didn't catch on in Europe where they were perceived as a singles band with more poppy tracks such as We're an American Band, Locomotion and Some Kind of Wonderful. You danced to Grand Funk songs at school discos, but you didn't ardently listen through their live double albums at home. No idea why. I knew them well from my days at the American School in Kinshasa where they were as popular as Deep Purple and Led Zep (and Sabbath).
Quo repeatedly tried to crack the US - more tours, more commercial producers, remixing their albums more "FM friendly" for the US market, none of it worked and by the late 70ies they had pretty much given up, a US remix of the Whatever You Want album being their last half-hearted attempt (it went nowhere in the US). At the same time they were hugely successful everywhere else (Europe, Japan, Australia), so eventually they did what GFR did and tour their tried and trusted markets. Noddy Holder of Slade (who also never cracked the US) once said that they shared the fate with Slade of "being too British" (together with the fact that they both shunned drawn out soloing which Holder felt was mandatory by audience expectations on the US concert circuit at the time: "We'd play 12 songs in 45 minutes and Humble Pie would play six songs in 2 1/2 hours, we'd watch them open-mouthed, it was a complete new world to us, but we just weren't that type of band."). But that theory can't account for the fact why a band so overtly Limey (and song oriented rather than solo-happy) as The Kinks always meant more in the US than in their home country.
And I really don't think that there is an American band that sounds remotely like Status Quo - that brash British, perennially throbbing and thrusting guitar sound coupled with the catchy melodies and their harmony vocals was pretty unique (if repetetive). For a band that did a lot of 12 bar influenced music, they were strangely unbluesy (as they were the first admit: they only began listening to blues in the late sixties, way after the British blues boom, the American blues greats meant nothing to them and they felt inadequate because of that). Foghat were way more bluesy than Quo, they were the white stadium version of black blues just as Rare Earth were the white stadium version of black soul.
With their penchant for songs in major keys, Quo are probably best comparable to US Southern Rock bands. Maybe the Lynyrd Skynyrds, Molly Hatchets, Outlaws, Blackfoots and 38 Specials of this world kept them out of the market, but then you could say that where there is room for a couple of bands of that type of music, shouldn't there always be room for one more?