For Bronxville & Scarsdale, you will need close to $2M - so your budget of $1M is not enough. For $1M, you need to look at New Rochelle (north end), Larchmont, Mamaroneck, Rye, Harrison, Pelham Manor, Tuckhaoe, Ardsley, Irvington.
As you learned, property taxes in NY is completely different than Calif. NY & NJ are the highest property tax states in the U.S. Basically, you're paying for top rated school districts & other services. The only consolation is if you're income is below $500,000, NY will give you a rebate for a portion of the school taxes paid. If you'e above $500K, sorry.