Blacklisting from the US, that is. Especially in comparison to other big British acts such as The Beatles or The Stones?
The Kinks landed in America for the first time in July 1965, after having had three top ten singles in the U.S. ('You Really Got Me', 'All Day And All Of The Night', and 'Tired Of Waiting For You'), each of which were particularly raw-sounding for pop in 1964, compared to the likes of the Beatles. The trip could potentially have been the Kinks' version of Beatlemania. Instead, it was something of a disaster trip where nothing was right. For starters, they discovered that they were billed as 'The Kings' in New York by the promoter. A little later in the tour they played a show in Springfield, Illinois where (they claim) the promoter was one John Wayne Gacy, who invited the band back to his home for drinks and got very upset when they decided to leave at 4am. In one small town, Ray Davies was set up in a room with a 14-year-old girl, with the sheriff waiting outside, and he had someone pull a pistol on him for being a longhair. Additionally, the band themselves were feuding, and the tour was haphazardly organised, leaving the band needing to drive long distances across America and then almost immediately play the show. All of this meant that the group was ...not on their best behaviour. Ray Davies punched a union official on the set of the Dick Clark Show. At a meet and greet with radio disc jockeys, according to Keith Altham, Davies called them "a bunch of that little word beginning with 'c', and it didn't go down well".
The aftermath of the tour, apparently, left them unofficially blacklisted; they were no longer able to tour, and radio was reluctant to play their music. A song like 'Waterloo Sunset' is era-defining in the UK; in the US, it never charted. The tour also left the band disillusioned about America; they went over expecting to hear Leadbelly on the radio, and discovered that American pop music outside of the British invasion was quite light and fluffy. They found, to their surprise, that America was behind the times, and that Americans by and large didn't appreciate the harder-edged American music that they loved.
While on respite in Torquay, England, recovering after the tour, Ray Davies wrote the song 'A Well-Respected Man'. At this point, it probably would not have been clear that the group had actually been blacklisted, but 'A Well-Respected Man' is very much the first Kinks song that's clearly in the Kinks' late 1960s vein, with melodies reminiscent of the (very English) music hall genre, and with lyrics that are very clearly about the British class structure:
And he plays at stocks and shares,
And he goes to the regatta,
And he adores the girl next door,
Cause hes dying to get at her,
The very clear Englishness of the lyrics are also present in the rhyming of 'regatta' and 'get at her'; in Davies' London accent, they rhyme perfectly, but most Americans singing 'get at her' would roll the 'r' in 'her'. 'A Well-Respected Man' was released in the UK on the EP Kwyet Kinks, and was quite successful; in the US, it also hit the charts, released as a single, it reached the top 20. Kwyet Kinks was meant to be a stop-gap acoustic EP, but it became the model for the band's new raison d'etre.
For British bands in the 1960s, America was the land of opportunity; America was the dream. With American cultural dominance increasing, British people who had grown up in post-war austerity looked on America and its dreamboat cars, its vitality and riches and its California sun, with envy. For the generation who became 1960s musicians - the Beatles, the Rolling Stones, the Kinks, the Animals, et al - American rock'n'roll of the late 1950s was an expression of the vitality of America and its freedom. So to go to America and discover the almost underground context of the music they loved (and the early Kinks were definitely fans of American R&B, covering the likes of Chuck Berry, Little Richard, Slim Harpo, and Bo Diddley early on) was eye-opening.
More so than the de facto blacklisting, I suspect the bad tour of America and their reassessment of America's coolness prompted the band's move to a more specifically English sound on 'A Well-Respected Man' and its ilk. The obvious influence on their post-1965 sound was the very British genre of 'music hall', as practiced by George Formby and his ilk (see Formby's 'The Window Cleaner'), which was full of cheeky British humour and playful, staccato vocal rhythms (and which is also obviously an influence on the likes of the Beatles' 'When I'm Sixty Four').
There was no particular reason why the Kinks' occasionally music-hall influenced music wouldn't be popular in the U.S. in the midst of the British Invasion; Herman's Hermits had a big hit with a cover of the old music hall song 'I'm Henry The Eighth I Am' in the US, to the extent that it became a plot point in the (very American) movie Ghost. And indeed 'Sunny Afternoon' was a top 20 single in the US in early 1966, despite however many DJs being unimpressed at Davies' foul-mouthed insults the previous year.
By 1966, the idea of 'Swinging London' had become mainstream, in the most obvious way - being featured on the cover of Time magazine. At heart, the idea of 'Swinging London' was the creeping realisation that London was more cool and hip and with it than New York or Los Angeles, something that the British had previously doubted (in an atmosphere, of course, that had long been dominated by post-war austerity and the distintegration of the British Empire). In this atmosphere, with London very much at the centre of the British music industry, pretty much every British band started to emphasise the Britishness of their sound; the Beatles in 1967 released a double a-side single, 'Penny Lane'/'Strawberry Fields Forever', with both singles being written (through some lysergic haze) about a Liverpool childhood ('Penny Lane' being about the main road in Liverpool that Paul would travel along in the bus on the way to school, and featuring a variety of very English characters). Even the Rolling Stones, in 1966-1967, got into the spirit of Swinging London, if somewhat begrudgingly - they were Americaphiles at heart. So you get the cod-classical piano, strings and hippie lyrics of 'She's A Rainbow', for example.
The sound of the Kinks circa 1966 very much fit into this milieu, and 'Waterloo Sunset' and its story of Terry and Julie and the Waterloo Bridge - 'as long as they gaze out on Waterloo sunset, they are in paradise' - is one of the defining British pop songs of the period. It was never a hit in the US. Its not being a hit in the US wasn't because of its sound - it's not that far away from contemporaneous folk-influenced American pop like The Grass Roots' 'Let's Live For Today'.
One of the most fascinating documents of the period, I think, is Ray Davies' review of the Beatles' 1966 album Revolver, in Disc And Music Echo Magazine in 1967; obviously, history has a better opinion of Revolver than Davies did at the time, but it's insightful into Davies' aesthetic at the time. For Davies, 'Yellow Submarine' - the most 'music hall' of the songs on the album - is "a load of rubbish, really. I take the mickey out of myself on the piano and play stuff like this." In contrast, 'Good Day Sunshine' is "back to the real old Beatles. I just don't think the fans like the newer electronic stuff. The Beatles are supposed to be like the boy next door only better." Davies is dismissive of anything basically rock'n'roll on the album - about 'Dr. Robert', he says "It's good--there's a 12-bar beat and bits in it that are clever. Not my sort of thing, though."
Where The Kinks' blacklisting, perhaps, plays a role in their sound is that in 1968-1969 or so, there was a movement amongst sixties rock'n'rollers to go back to basics - to ditch all the psychedelic, electronic stuff, and just make good old rock'n'roll (which was nonetheless quite markedly different to the rock'n'roll music they made in 1964). So you get the Beatles doing songs like 'Don't Let Me Down', the Rolling Stones doing 'Jumpin' Jack Flash', the Beach Boys doing 'Darlin', etc. The Kinks didn't follow this back to basics trend, which was the source of some discontent in the band, some of whom were sick of sweet ballads. Instead in November 1968 they released the album Village Green Preservation Society, a concept album celebrating a sort of mythical English 'village green' (e.g., 'Village Green'), still very much in the folky 'Waterloo Sunset' vein; 'Village Green' has Nicky Hopkins playing cod-baroque harpischord, and has a prominent oboe part.
The Kinks eventually toured America again in 1969, but they basically stayed in the world of very English concept albums for a decade - 1971's Muswell Hillbillies, Preservation Act 1 and Preservation Act 2 in the mid seventies, etc - until the late 1970s, when on albums like Low Budget and Misfits they went back to a more straight-forward rock sound (which very self-consciously harks back to their early rock sound on a track like 'Destroyer').
Sources: Rob Jovanovic, God Save The Kinks: A Biography (for specific Kinks stuff), Yeah Yeah Yeah by Bob Stanley, and Tune In by Mark Lewisohn (for context)