In terms of public funds for this - note how different this would be compared to what Vegas is doing or what Tampa Bay is doing. Those areas are paying to keep a team there, maybe not in an ideal spot, because they want to use it as a way to make their city seem important. This is also why the state spent money on the White Sox's park 30 years ago, and why the City paid to renovate Soldier Field. These are the projects that are big money losers for cities.
If Chicago did this exact project with the White Sox, it is fundamentally different. This is an urban renewal project. This area is blighted, it has been undeveloped for more than 50 years. Between this site, the rail yard, and the river, it creates a large area of the city that is unwalkable and a barrier to high density development. It is also basically paying no tax to the city, it probably creates costs to police it, and it will be a drag on the surrounding neighborhoods.
If the city were to find a tenant to develop this area, it might take more than 20 years to recoup any funds they invest, but if the city could turn this into a developed area - they can think about returns in the 50 to 100 year horizon.
Furthermore, developing this site will allow denser development around it. Removing the blighted area in this spot will increase land values nearby and lead to the potential for significant development in this area and moving farther outwards.
That's all true for literally any tenant here, which is why the city would likely put up some significant money for any development in this area. However, there have been several attempts to develop this region, all of which, so far, have completely failed. Other developers have different risks - the office market is currently pretty weak following COVID, so if the most recent developer was planning to add significant office space here, there's likely no funds available for that right now.
For the city, the benefit of doing this with the White Sox is - the White Sox, as a baseball team, are likely to still exist in 30 years. Their funding situation and ability to raise funds is different from a business relying on other real estate markets. If they can get the White Sox on board as a key tenant here, and the city puts up some money, they will actually get the site developed. The White Sox aren't likely to cease to exist in 2027. This isn't a real estate developer announcing the project and then hoping to get funds lined up once they show interest from enough businesses.
The White Sox building a high quality park here that draws 25,000 a night 81 times a year should also lead to additional business around the area. It won't take long before someone opens "Big Hurt Bar" across the street, hell I'd do that if I could. Add a hotel on the site, and you've now taken a blighted, undeveloped area and turned it into a center of activity and development in the city.
The city should not pay the entire project, the White Sox should be putting up a significant amount of money as well. However, note the difference between rebuilding Soldier Field or building New Comiskey. In those cases, they were just spending money to keep the named thing in the city. Soldier field kept the Bears in the city, but spurred likely very little new development. New Comiskey gave Reinsdorf his parking lots, but I don't think there's any businesses in the area that rely on or cater to the White Sox. Hell, I think "Jimbo's bar" was the spot they showed on TV during the last game of the 05 World Series and that closed a couple years later. When the new ballpark was built at 35th street, people arrived on the highway or rails, went to the game, and left - it supported basically no other business. A walkable stadium, with businesses nearby, a hotel and other development on the site, with transit options as the main way to bring people in - this supports the entire area.
It is possible they can't make this work, the two sides will need to agree on how much money each side is going to put in and that will always be complex. But this is fundamentally different from just building a new park to avoid losing the franchise - this is a redevelopment project. Any developer on this site would expect a significant contribution from the city to bring it back to the tax base, and the White Sox offer substantial benefits compared to your average developer. It is to the taxpayer's benefit to come up with an agreement and get this done.