Seems like that would hold true to its original purpose of not taxing someone out of their home, make investment in property somewhat less appealing, and provide a more predictable funding source for schools.
Too, the policy that reassesses the site upon re-sale takes away mobility for Bay Area residents-- this lock-in is extremely destructive for the housing stock (not to mention transportation), and will continue as long as Prop 13 exists in any form.
The idea that "being taxed out of your home" is a bad thing is extremely overblown (remember that any person to suffer this malady is suffering from overly valuable land). Rather than the heavily distortionary effects of Prop 13, better approaches would be:
* Building more. That's what the folks who suffered high property taxes in the 1970s should have done. The property tax burden for each housing unit would be driven down.
* Community land trusts offer the possibility of stability for residents who want to be completely removed from the rises and falls in housing value.
* For the "poor widow" situation, a direct subsidy on an as-needed basis (food stamps for property taxes, as it were) is a better solution than what Prop 13 does.