I’m not saying this is the best metric but they would find that sort of treatment to be highly ineffective. Lifetime prevalence of depression and the depression at any given time are different things; it would be like trying to weed out all future cases of flu based on symptoms at hire time. Either way it wouldn’t make a dent on flu caused by the work environment itself.
It's a simple solution, but a bad one. The alternative is to overwhelm people with complete availability and encouragement of help, and while helping them, convince them that it's not in their best interest to own guns (ideally to agree to give up their right to own guns.) Actually effective change is expensive, so will not be done.
That being said, tracking tech suicides doesn't have to track the companies they work for in order to indicate if there's something wrong with the culture. Theoretically, identification of companies would certainly help to figure out what that wrong thing is, though the metric would immediately be gamed in the way you describe and exacerbate the situation.