If property taxes increased substantially, I’d have had to raise rent...
What happens when you increase property taxes is that property values go down to compensate; it's less valuable to own when you have to pay more to do it. So in the short term the property tax increase hits whoever owned the property when the rate change was announced. But then it becomes less profitable to own property in that area, so people invest less in building new housing there, or even in maintaining the existing housing. That tends to constrain supply and cause rents to increase until you're back at equilibrium. The long-term equilibrium is based on the relationship between rents and total costs. Property tax is a cost so in the long-term higher property taxes result in higher rents. (This isn't counting the pathological cases where higher property taxes reduce property values and investment and the government responds by further raising property taxes until the locality is in a death spiral. Naturally rents in dying localities tend to go down.)
In California long time landlords get Prop 13 tax cuts. New ones don't. They compete for the same tenants and charge the same rent.
Long time landlords have much bigger profit margins because of that law.
Other states have property tax restrictions but I don't know as much about them.
"Long time landlords" and "new landlords" are not inherently different people, they're the same people at different points in time. It's like saying a landlord which has paid off the mortgage on their building makes more money because they don't have to make mortgage payments. The value of the future reduction in property taxes gets priced into the purchase -- but then so does the cost of higher property taxes at the beginning, since the government still needs a given amount of revenue.
The thing Prop 13 really does is to discourage long-time landlords from selling the property at all, because the buyer will have to pay higher property tax than the seller which makes the property "worth more" to the seller, even if it's otherwise worth more to the buyer. For example, this can discourage development because the existing owner doesn't have the inclination or the capital for expansion but would demand enough for the property that it would make it unprofitable to a third party who would have to pay higher taxes.
With the way you can inherit property and keep its valuation, there may be a class of "long-time landlords"-- individuals and organizations-- emerging in major cities. Especially, as you note, since Prop 13 discourages transactions.
edit: YES, it's a nice plan to just raise rent to cover the school taxes. However, unless you are in a popular market, your rent prices may not bear the increase. Example Northeastern PA has cheap rent because there are so many places available, and thus the $4000/yr in school tax portion of the property tax is not able to be passed on to renters.