If parents have been able to afford private schools for their children prior to this then they can afford additional tuition & extra curricular activities when moving their children to state school.
Plenty of children move schools at inopportune times, even mid-term, and turn out fine.
As with any paid service, one should always be prepared for price increases. Capitalism does love them. If parents can only just afford the fees of independent schools without rises, then they take the risk that they will have to withdraw their children should the fees go up more than they can afford.
When balancing this against the injustice of tax breaks for fee-paying schools, I really don't see the problem. As with all fiscal decisions there will be some people who fall through the gaps, it's unfortunate but that's they way it is. Sounds brutal? I wonder how many parents of privately educated children protested all the many policies of governments over the last 15 years which have at best disadvantaged and at worst actively harmed, even killed, many others.
It's not that I don't care about your children, but I don't care as much as you do how this decision will affect them. They will not be plunged into a lifetime of hardship as a result of this. They have supportive families who are able to spend many thousands on securing for them advantages others don't have. It isn't class bitterness (I'm well educated, middle class, white) - your children will be OK.
I feel much for SEN children who attend fee paying schools because there is quite literally no other education for them, but this is a policy which is long overdue and needs to happen.
Independent schools are businesses. If they're liable to fold because of a removed tax break then they're not particularly well run businesses. If your child will not be able to access their services any longer, that's really not the fault of the government.