I wouldn't call it an unsolved problem necessarily. We probably disrupt the ecosystem more when we build a parking lot.
We're taking the salt out of the water in order to "drink" it, but the reality is that most of it gets used for irrigation or toilets or whatever, and eventually returned either as groundwater or to the ocean.
Obviously it depends on the location of the sewerage outflow relative to the location of the plant.
Of course this only matters at scale. For the suitcase sized device mentioned in the article the amount of brine generated is negligible.
The point is, people are always writing letters to the editor and what-not when some green tech has some environmental impact, far more than they do over their city making another parking lot or any of a million other things. I assume it's because it triggers some notion of hypocrisy. c.f. wind turbines and dead birds.
How about we put our energy into fighting cars and parking lots first, before we worry about solar-powered desalination that could potentially save the Western states from the over-extraction of water that's turning them into dust bowls?
We really should have more unpaved outdoor that is simply trimmed semi-regularly if or when the plant growth gets too big.
[1] Unless you're in a sandy or snowy area, where shoveling the sand or snow is difficult on an unpaved surface.
On my driveway we only tend to get clovers and really resilient weeds. So, either really shallow rooted plants or weeds with absurdly strong roots. You're not likely to get native plans to grow unless you section off parts of the area for maintaining living soil.