What a lot of hypocritical patronising rot from some of the posters on here!
Paying upwards of £25,000 per year for private education, excessive tutoring and paying to be in the catchment area of a great school are just other means by which parents buy advantage for applicants.
You are doing nothing worse than this OP, you're just doing it a bit too late in the day, rather than from the age of five like a lot of them will have done.
Anyone would think their little darlings had dropped out of the womb spouting Proust, such is the contempt for more obvious forms of preparation.
The top up prep you are considering is no less vulgar than paying for prep school - the clue is in the name there!
Some of the candidates applying for Oxbridge will have had years of debating prep, so when it comes to thinking "on their feet" the replies will naturally appear to be spontaneous and unrehearsed.
And don't tell me that the academics have some form of supernatural ability to distinguish between merit and polish.
I certainly think not!
Potential is laid down in the genes at birth - all the rest is polish of one form or another.
Some folk have the greater means with which to add more shine.