I’d like to thank San Marcos and warmtotheidea for their comments.
Warmtotheidea: Indeed artificial turf gets very hot especially in the summer early afternoon. Our model computed a temperature of 122 degrees farenheit in a suburban setting, but on more open surface like athletic fields and further away from the coast I would estimate the temperature to be up to 20 degrees higher. I would say that if you plan a surface to sit on (like in a park) I would not use artificial turf. However, for surfaces that have mostly foot traffic — athletic fields, landscaping around commercial buildings, dogparks, residential front yards — the hot temperature will not be an issue. It is no hotter than asphalt so don’t expect your shoes to melt.
San Marcos: You brought up a great point that I should have discussed in more detail. You are right that plants take up the greenhouse gas carbon dioxide convert it and store carbon in plant material. However, after the plant dies (or is burnt), the plant material decomposes and forms carbon dioxide again (to complete the cycle), or even the more damaging greenhouse gas methane (e.g. in landfills). For trees indeed the carbon dioxide will be removed from the atmosphere for a long time. Also if you grow plants to make ethanol, or burn plants in a power plant to create electricity, CO2 is produced, but that is offset by avoiding CO2 from oil, coal, and so on. In these cases the CO2 taken up by the plant indeed reduces or delays global warming.
Grass, however, is mowed frequently and — to be best of my knowledge — no energy is extracted from the mowed substance. Rather it decomposes releasing the CO2 that had just been taken up a few weeks earlier back into the atmosphere. So grass does not contribute to a reduction in global warming.
Regarding your other point on storing water. Unfortunately once irrigation water has been applied it is impossible to recapture it, since the plants will remove it from the soil as water vapor. Only when over-irrigating onto sidewalks, streets, one could in theory recapture that water, but it would be much cheaper and easier to improve the irrigation system.
I agree with you that there are more sustainable ways to manage grass without fertilizer, using human-powered mowers, letting it grow taller to reduce evapotranspiration, capture shower water for irrigation etc., but I think we can agree that no more than a fraction of a percent of the population is willing to do that.