There was a brief period at Wembley where we scraped passed a few teams without Kane - somehow it was decided we were better without him.
This is right. The "better without Kane" meme comes from a combination of failing to adequately consider the context of the matches without him and misapplying statistics (which is understandable because the vast majority of people don't have, and can't really be expected to have, a solid grip on university level stats).
The context (at least as far as the '18/19 season goes) is one of cognitive disconnect between the raw results and the performances. In short: last season, Spurs won a lot of games without Kane where victory was probabilistically unlikely ("lucky" -- as much as I hate that word) and lost or drew a lot games with Kane which should've yielded higher average points. The raw results were exceptionally unflattering to Kane.
xG is a good benchmark for individual shots (which is, after all, its intended purpose) and also fairly good for assessing the performance of a player or team over a long series of games (when summed). xG is terrible for deciding which team "deserved" to win an individual match for the same reason that buying 10 lottery tickets gives you a poor chance of winning compared to someone with more than 2 million tickets, or even just a few hundred.
When you sum xG and xA differentials over the "with Kane" and "without Kane" conditions from last season, you find that - insofar as these particular models can be used to derive such conclusions - the team were far more likely to win the average game with Kane than without. The blocks of games where Kane was injured just happened to coincide with periods where the team was most likely to do extremely well versus their xG/xA. Sure, players like Son can generally solidly outperform their xG over a long period of time, but that doesn't explain much of the variance at all. Kane has one of the highest mean xG differentials in Europe since '14/15, which is consistent with him being one of the best strikers of a ball in modern football. If Kane had been playing in the games he missed, it's likely that Spurs' GF would've been even more emphatic.
The stats part is really - well, by far the hardest bit is wrapping your head around the stuff above! - to do with the relatively small sample size of games without Kane. Alone, the sample with the particular results isn't big enough to approach statistical significance. Worse, the massive difference in sample sizes between "with" and "without" makes them difficult to compare in any meaningful way. Many statistical tests prefer at least sanely balanced sample sizes.
Frankly, though, you don't need to be able to understand any of the above to put the hypothesis to a simple eye test. If you go back and watch some of the games without Kane, you'll see plenty of turgid football followed by a chain of fortunate last-gasp goals for narrow-margin victories. After that, go back and watch the first few games from when Kane returned the first time: observe how he still scored in consecutive games, with a lovely solo goal on his return against Burnley, and observe how the defence then ship enough at the other end to render Kane's goals meaningless -- twice in a row. I think it's particularly instructive how visibly the energy levels from the other attackers drop after Kane comes back against Burnley: as if they know they have to step it up in his absence before falling back to "allow" him to bear most of the responsibility again when he returns.
So even though Kane was doing his job and scoring last season - sometimes creating for himself out of nothing much, a la Burnley - it was very easy to simply conclude "we were winning when Kane was gone and now he's back and we're not winning". It was the most superficial analysis possible, and, unfortunately, because most football fans on e.g. Twitter aren't capable of much more, that was the analysis that stuck and became a meme.
The media, too, was complicit, whether through sloth, greed or actual incompetence (I suspect all three). Supposedly reputable sites like Sky Sports provided column space to pundits who parrotted the surface-level analysis and posted articles full of misleading stats, such as Spurs' overall five-year win-rate with and without Kane (useless for reasons already discussed; you can use this to make it look like Kane makes almost no difference to Spurs when he was individually responsible for winning more points than any other player in the league in '17/18).
The new ice cold take I've spotted on Twitter is to blame Jose because "Poch could get the team winning without Kane", which ignores everything I've said. Sure, the team have proven that they can do as well without Kane or even luck on their side in isolated patches, but the proposition is entirely untested in the mid-to-long-term. I suspect that a lot of fans are going to be shaken out of their stupour of taking Kane for granted over the next few months.
Bad statistics in the wrong hands, "social" media memes and not thinking critically for yourself form an incredibly dangerous combination. Luckily, in this case, it's only football. Less fortunately, the exact same thing is ubiquitous in real life.