Talkingnonsense - yes, that's probably a factor. I certainly think there's a lot of lazy presumption in the way people automatically associate "academic involvement" of parents with "privilege".
In fact I dispute the idea that the existence of tutoring automatically creates clear immovable categories of haves and have nots. Around here, tutors generally cost about £30 an hour. We had one for our DC, once a week for two terms before the 11+ (although they weren't much good and I ended up doing most of the work myself). At say 12 weeks per term, that's 24 x 30 = £720, total.
Is that an amount that will be unaffordable for many people with children on FSM? Yes, probably (the FSM threshold is extremely low and not generally accepted as a very accurate indicator of class or disadvantage, since you can be really quite poor and still not under it). Is it, WITHIN THE GREATER SCHEME OF WHAT IT COSTS TO RAISE A CHILD ANYWAY, an amount that automatically excludes all working class children and confines 11+ success to the "privileged" few? Nah, that's bollox.
In fact there are plenty of people with less of an educational, professional culture than my family has, who earn more money than us. (Seen what plumbers charge these days?) Many of these people don't tutor their kids for 11+, but it's not because we can afford it and they can't. It's just because it's not part of the way they think or what it would occur to them to spend their money on.
Now of course I'm aware that there are areas where tutors cost more than that, and superselectives where candidates are tutored for longer than that. OTOH there are also plenty of people in our area whose kids pass 11+ with no tutoring at all - the claims that that's impossible in fully selective areas are simply untrue. Partly of course this is because so much of the difference in a child's academic orientation is already created by that age in so many other, broader ways - whether their parents read to them when they were little, whether their parents TALKED to them when they were little, etc. etc. So it may be that tutoring, while high profile and obvious and an effective way for parents to spend their money reassuring themselves that they're doing all they can, isn't actually such a major factor in determining success.
Aren't there some grammar schools (eg Tiffins, I think?) that go out of their way to make their entrance exam as unpreparable as possible? That probably makes little difference though as the kids are still affected by the amount of input and support that has gone into their education (in the widest sense) generally up to that point.