A better argument is "distributing the tax burden based on how long somebody's family has owned property in the same place has a large number of negative effect

A better argument is "distributing the tax burden based on how long somebody's family has owned property in the same place has a large number of negative effects." One (of many) serious problem(s) with Prop 13 is that it removes an important feedback mechanism whereby inflating property values result in higher taxes which gives even those who currently own property a financial incentive to support policies that keep housing prices in check. With that feedback mechanism removed, a property-owning homo economicus's rational move is to support policies that drive property values sky high (e.g. very restrictive residential zoning to limit competition while simultaneously seeking intense commercial growth to boost demand) at the expense of other members of the community (e.g. renters and future generations), and that's exactly what has happened in California over the last several decades.