OP you may be right that 11+ is unfair and elitist but none of your arguments appears to support your conclusion.
- it favours children who are well-rounded, and who are so at 11ish. A child who develops well academically but later, and/or who is stronger in the verbal part than the non-verbal, or vice versa, won't do well
People's academic abilities develop at different pace throughout education, from GCSEs to postgraduate studies, so this is an issue with all testing, not just 11+ (so if you are right all tests are unfair - a reductio). Moreover, most people have strengths in different areas, rather than being equally good in everything. Besides, a child can enter even in a super-selective school by acing one subject (english and verbal) and doing OK in the other (Maths/NVR). One may say that the difference in development can be starker at 10. But there is an official adjustment for summer born kids, who develop later. In any case, this point does not relate to your charge of elitism.
- state schools do not typically prepare children for these kinds of tests, so the family situation becomes a huge differentiator: if your parents are more educated, and/or take you to the library, and/or can pay for tutoring, you'll have a huge advantage. Libraries have books to prepare for the test, but a teenager can go to the library alone, not a 10-year old.
This is the main argument, but as others have said, it is family/parental ambition that is the main driver behind academic success at 11+ and this cuts across middle class/working class divide. Hence so many grammar schools are full of kids of immigrants from disadvantaged backgrounds. Now, you may say that family ambition, be it by rich or poor/disadvantaged families, is itself an unfair factor. That's plausible, but again it is not elitist, in a class-based sense (understood as either income or culture).
- some of the verbal part is honestly too hard for a child of this age. I am not sure it is appropriate to expect that 10-11 year olds know vocabulary such as cantankerous, recalcitrant, cogitations, etc
That's really irrelevant. In a super-selective grammar, the top (say 240) ranked kids in the exam will be admitted. The harder the test, the better you can differentiate numerically between the exam performances. If the test was easier (so that all kids would know the vocabulary) then you would get 400 kids with identical scores. The hardness of the test is not elitism, it is a way to distinguish between the thousands of applicants, hundreds of whom will be very strong and bright. Schools don't set out to test vocabulary that a 10 year-old is not supposed to know. In an exam with 300 applicants for 240 places, the fact that most kids don't know cantankerous would be irrelevant, because they would get a place.
- children who speak a Latin language (maybe also Greek? Not sure) have a huge advantage guessing the meaning of the more complex words. French-speaking, Spanish-speaking kids etc are much more likely to guess the meaning of initiate, abound etc even if they are not avid readers
The studies of which I know, suggest that bilingualism confers an advantage in late teens and onwards, and it is a disadvantage early on (at primary education). But even if you are right, this again cuts across class divisions. Even if unfair, it's not elitist. But I struggle to see how language competence at 10 yo, enhanced by the fact of living in a bilingual household, is unfair. You contrast this to being an avid reader. Why is living in a household where one is encouraged to be an avid reader fair, but living in a household where one speaks many languages unfair. Unless someone assumes that only innate intelligence/ability should matter, and that any cultivation of skills by family background is unfair. But then this would suggest that we should test even earlier. Besides the assumption, made by the 'father' of the 11+ that intelligence is entirely innate, is flawed.