Even if you could do that, it wouldn't work. It would simply make all of the poor neighborhoods more attractive to gentrifiers at the same time.

I think there is a tipping point where if you help poor neighborhoods too much, they become attractive enough as places to live that the poor residents end up being displaced by increased housing costs. It's an unintended consequence.

The difference I see here is that the developers and new residents are the ones who reap the benefits of the increased property values from gentrification, whereas if the city makes modest improvements before it's current residents are forced out by increasing prices/taxes, then the current residents reap the increased value. Even if they move because they don't want to pay taxes on higher property values, their property is still worth more when they sell it.

Typically with gentrification it isn't a series of small improvements that change the neighborhood its bigger urban planning changes or additional big projects.