So Fort Worth has one plaza and therefore a second one might be "too much". I suppose I will have to chew on that one for a while longer.
In the meantime, here is a square in a well known city that begs the notion of whether there can be too much open space:
You are misinterpreting me.
An urban park or plaza isn't just about having a nice space to be - it's also about how it fits into its context. There needs to be density and activity and activation for the space to play off of and work with. It can absolutely be harmful to put too much "open space" into a city if it's not planned properly and isn't playing off active, engaging buildings. The parks and plazas work only if the city is dense and lively around them, otherwise it's just like so many existing under-used "open space" that already exists in cities like Fort Worth.
It's important that downtown Fort Worth fill out a lot of these blank, empty blocks with density, not just more parks and plazas. Burnett Park is example of how under-utilized a space can be when it's just surrounded by office buildings with no real interaction with the public realm. One can say it gives Burnett Plaza office workers a place to smoke or take a break, or whatever, but so would a series of sidewalk cafes along more engaging buildings and a smaller, more energetic plaza or just nice, well-designed sidewalks.
(Also, downtown Fort Worth doesn't have "one plaza" - it has several public plazas and parks, almost all of them apart from Sundance's having problems with under-utilization due to being tied to un-engaging, deadening contexts. This just drives my point further home.)
One could absolutely design a public plaza that would work on the Landmark lot, but I don't believe that just plunking one down because "open space" and "we want to look at the Fort Worth Club building" is a recipe for a truly sustainable, healthy public space. A lot of careful thought and planning would need to go into it so that it doesn't just become another Burnett Park, a ghost town 95% of the time.
I've seen both Chicago and Portland brought up in this thread as examples of places with lots of public spaces, which again just furthers my point. I don't think I need to point out how much more dense and activated Chicago is than downtown Fort Worth, but even downtown Portland's public spaces work because they're playing off a much more dense and engaging context than downtown Fort Worth.