It's a complete throw of the dice ime!
Here's my anecdotal contribution...
DS, summer born, had a serious speech delay, by age 2.5 had exactly 5 words he could say, by the time he was due to start school he was clearly lagging behind his peers in everything, including physical skills, fine and gross motor, but the worst part for me was his social skills were really behind because the speech delay put him on the fringes of everything that went on.
I was in one of those areas a pp mentioned where postponing school meant you started in year 1 instead of reception, so no actual delay in practice, just joining into the same year but without the relationships everyone else has by then. Ridiculous system!
So not only was he summer born but he was behind in milestones etc.
Taking him in on that first day felt like feeding him to the lions. I hated it. I know it was sink or swim, and I greatly feared it would be sink!
His learning curve was super steep but after one year you couldn't tell him apart from others so easily. By the time secondary came around he was itching to move up and out of primary school as he was academically and socially ready (amazing to me, looking back!). In his peer group he is standing out in confidence and academics. I could never have predicted that for him personally when I had that struggling 3yo kid.
So the statistics and how he was at age 3 all pointed to keeping him back, and I would have done if our area allowed it (sensibly).
In contrast, his October born sister seemed well ready for primary school. One of the oldest in her year, confident, articulate bright and happy and hitting all her milestones with ease she skipped on in to school, while I cheerily waved her off, happy she was embarking on a fun new adventure.
As time went on however, she didn't do so well, struggled with lots of things emotionally and academically, she is currently under assessment for autism and needs a lot more scaffolding to support her schooling experience, though she's doing great and is happy.
I'm very grateful she is October born as she has matured much more slowly and has need that extra time as she is young for her age.
So, the stats can bear absolutely no relation to your child.
Your child at age 3 present very very differently to how they are age 10.
A decision to keep them back could be the swing thing entirely and your child could be cursing you, or out could be the right thing.
My advice would be to put that stats out of mind and base your decision on the direct info you have available, including family logistics! Another year of nursery is expensive! An additional child at home might benefit from some 1-2-1, if this year is a high/low birth year will you be more or less likely to get your preferred school of you go now/wait...
Factor in all in... Then throw your dice and in 7 years time you'll know if it was the best decision. 😁😁
Either way, you'll all survive it.