I don't know that the voucher system would overcrowd schools, as much as eliminate the multiple levels of administrators.
Lets take Cheektowaga as an example. Typical suburban town of 100k people. They have 7 school districts. Seven! Cheektowaga Central, Cheektowaga Sloan, Maryvale, Depew, Cleveland Hill, JFK and Lancaster. Completely ridiculous.
Lets say there was a voucher system and all the kids wanted to go to the "better" schools, Maryvale, Lancaster and Depew (I don't really know which are the better ones, just making an example). In order for those three to maintain their "better" status, they will have to hire more teachers to maintain student to teacher ratio. The schools not getting the vouchers would eventually close, leaving many highly paid administrators out of work, but the demand for teachers would remain constant.
The only thing is you would have a bigger school.
As far as gov't $$ going to religious schools, all religious schools take some level of gov't $$ in order to comply with gov't mandates to keep accreditation. The voucher system would not apply to private schools, only public schools.