What is your basis for believing that governments believe this? One of the interesting lessons from the disastrous Mark Latham experiment was that there are many votes to be lost in cutting govt funds to private schools, even amongst those who are unlikely to be able to send their children to one. That appears to be indicative that governments fund private schools for political reasons rather than for economic ones.
Governments fund things almost entirely for political reasons, or else they wouldn't get elected. You should be asking yourself why it is parents don't want to send their kids to public schools in the first place.
This assertion is often made by supporters of taxpayer funding for private schools, but it is just supposition, which may or may not be correct. It is easy to draw erroneous conclusions by comparing the unadjusted govt funding provided to private schools with the average budget per pupil in public schools, but that is no indicator of what the average cost would be if all students were in government schools (given that private schools would then no longer be able to scoop off the cream of the socio-economically advantaged students who are much cheaper and easier to teach, and that the public system would have increased economies of scale), and even less indication of what the cost would be in the much more likely, and less draconian, scenario in which private education was permitted, but simply did not receive any taxpayer funds.
The benefit of having a favourable economy of scale is negated by the public system being overly bureaucratic and government run. Again, look at the BER program where, in theory, government run schools would have a much better economy of scale (being able to negotiate larger contracts for more work due to much of the work being commissioned by government departments responsible for hundreds of schools) ending up being vastly less efficient than independent private schools at allocating funds, who theoretically should have had higher costs.
If you believe that public schooling does as well, or better, than private schooling on a dollar for dollar basis, then you would have no problem supporting a voucher system in which the government provides a fixed value voucher which is attached to the student. As with most (if not all) voucher systems, rules would require that schools not be able to accept a voucher if they charged school fees on top of the voucher, thus preventing the wealthy from using it as upper-class welfare.