Every normal person wants to do the best for their children.
FWIW I know people who have only had one child, and still lived in a cramped one bed flat in order to send their daughter to a decent private school. That is sacrifice.
I also know others who have mortgaged to the hilt to enable the same education.
If you can afford it, then do it. The benefits far outweigh the disadvantages. The contacts you make generally last a life time and the world may have moved on, but it's never going to change to get rid of nepotism and the who-you-know buddy system.
Re mortage interest rates - Bank of England forecasts they will remain this low for 5 years.
In your shoes - private prep then into grammar, but you have to be in a grammar area of course. But be aware that a private education doesnt buy brains, and if your child isn't performing you will be out the door, prep school need 100% pass rate, or near as damn it. You are there by invitation, not by right. I've seen a few manouvered out in Y4 or 5 when it's obvious they will screw the statistics. Also be mindful that prep schools can be bankrupted and close, as has happened locally - parents cant afford to pay and suddenly you have a whole heap of children scrabbling round the borough trying to get into the better state schools, which are over subscribed anyway.