I stumbled upon an article by the guys at The Athletic earlier. The gist of it is that Kane was not for sale, full stop. It's a long read so I'm only posting a couple of excerpts.
Optus Sport
sport.optus.com.au
First excerpt.
But the basic facts of the saga are as follows:
Second excerpt.
- Levy was furious with the way Kane tried to bounce him into the move at the end of last season.
- City offered five players in May and £75 million plus £25million of add-ons in June, then made no further bids.
- They were prepared to pay a fixed £100 million plus £20-or-so million in add-ons, but Levy refused to negotiate.
- Spurs fined Kane two weeks’ wages for delaying his return to pre-season training.
- With the two clubs not talking since late July, the move had been dead for weeks despite Kane’s hopes.
If this was a defeat for Kane, it is also a win for Levy. He had been insistent all summer that Kane would not be sold, especially not to another Premier League club. Those close to Levy have always known that this was not a negotiating position, but a firm commitment.
Part of Levy’s position was coldly rational. He knew this would be a bad market in which to sell a top player, and that Tottenham would always be better off keeping Kane for at least one more year. By the end of next season, they would either have improved their situation enough to convince Kane to stay long-term, or they could sell him in a more buoyant market for at least as much as they were offered in 2021. The fact that City’s only cash bid for Kane was £75 million up front with another £25 million of add-ons has effectively proven Levy right. There was nothing to be gained from selling the striker this summer.
The other part of Levy’s insistence on keeping Kane was personal. He felt deeply affronted by the conduct of Kane and his family throughout this process, from the leaking of Kane’s desire to leave and that Neville interview right at the climax of Spurs’ season in late May, through to Kane’s unauthorised absence from pre-season training at the start of this month (which cost him two weeks’ wages). Levy saw this as a challenge to his authority, which made him even more keen not to get pushed around. Maybe with a more delicate approach, Kane might have succeeded, but Levy was insistent from early on that he would not be giving in.
Hard to imagine in all you research of this topic, you did not stumble across the hundreds of articles back in the summer that said part of the reason talks broke down was... "Conte wanted a guarantee that Harry Kane would not be sold"

Antonio Conte issues transfer demand before accepting Tottenham job
Antonio Conte wants Harry Kane future assurance before taking Tottenham job According to The Times (h/t Express), Tottenham Hotspur’s pursuit of Antonio Conte as the next manager has taken a fresh twist. The Italian has issued one condition that needs to be met before he accepts the job. Talks...


Tottenham twist as Conte ‘demands Harry Kane stay before taking job'
Tottenham want to name Antonio Conte as their permanent replacement for Jose Mourinho.

Tottenham Opinion: One of Antonio Conte’s stipulations cannot be guaranteed
Antonio Conte is on the verge of becoming Tottenham's manager, but his main stipulation, that Harry Kane not be sold, cannot be guaranteed.

Antonio Conte wants the Tottenham Hotspur’s head coach job – as long as Harry Kane stays
Tottenham Hotspur have had talks with Antonio Conte but there are stumbling blocks if he is to be appointed their head coach.The Italian left Inter Milan by mut

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IF Kane was to be sold, however, Conte wanted guarantees that at least part of the fee generated from the sale - likely to be in excess of £100 million ($141m) - would have been used to sign an adequate replacement.

Conte talks hit roadblock as Tottenham press on with Paratici signing | Goal.com UK
Tottenham’s hopes of appointing Antonio Conte as manager are all but over after talks stalled over a number of key issues, Goal Italy can confirm.
www.goal.com
We'll never know as the press is not exactly the best source of info, is it?
But I'm curious as to why you left these easy to find articles out... Did they not fit your narrative?