My understanding is that the greatest barrier to expanding heritage protections is the property owners' choices. The city can decree that a certain property is a Municipal Historic Resource against the owner's will, but what I've heard is that they then have to compensate the owner for any potential lost value. In this sense, the budget is a constraint, but I think the expenses for contributing to keeping up the historic character of voluntarily designated properties are relatively modest.
I'm all for expanding heritage protections, but we should be careful about what we advocate for. Just because a house is old and it's a 'named' house (i.e. we know who lived there in like, 1920), doesn't mean it's historically or architecturally important. I don't think our goal should be to keep entire neighborhoods frozen in time.