Not true—at least didn’t used to be. For 3 years, Tesco renewal was cheaper for us than any online comparison site (same excess, options etc). That included Tesco new quote.
But definitely never take a renewal without checking.
Funny thing was it was a Tesco policy that doubled the price, though for the previous two years they were either the cheapest or so close that I didn't bother changing; I think last year there was one policy that was £15 a year cheaper. This year though they were far from cheapest.
Yes, all bets are off this year. Plus we now have a ‘claim’ even though it was all paid by the other party. No idea how much difference things like that make but certainly not comparing like with like.