Economics is all about the choices people make, and the IFS have very obviously misunderstood (or not really thought about) what's driving people's choices. Especially the people who are at the boundary of affording private school, and so may be influenced to stop or else not to start. That means they are wildly off the mark on what they'll spend that money on instead.
Eg If you increased the cost of TVs, then buying the TV is now worth less to the buyer than the same cost of other consumer goods. For that higher price, they could buy a games console instead (and they prefer that) so that's the choice they make.
But people don't buy private education as one of many possible consumer goods. Because it isn't a consumer good.
Buying cars, bikes, clothes, furniture, electronics, sports equipment: it's simply not the same spending category.
The equivalent things people will choose are all ways of improving your family and personal life. You're not going to replace giving your kids the very best education you can with a games console or a car. You're going to replace it with dropping a day of work or retiring earlier.
Holidays and family outings and activities probably do fall in the 'improving family life' category for some families, but not to anything like the same amount of spending. (There's a concept called 'marginal value' in economics, which is the value buying a bit more of something has to you. You have to spend the full £20k on education if you want it at all, but you're not going to spend that much on holidays since at a certain point spending more in holidays is worth less to you than spending an extra day a week with your kids or retiring earlier)
It won't be obvious though, so I expect we'll never know. UK productivity (and hence everyones standard of living) will just continue to drop.