Those are naive arguments. Most people
need employment, and most people want to get better wages by climbing the ladder.
Get it in writting? When in the history of the mankind it ever happened? Not to anyone I know, and I would never make such promises as well. Promotion depends on a lot of things, and promising something like that in writing is stupid.
And once you're tagged as someone who might be leaving, some employers really may prefer no to invest in you. Why should they? You might leave.
The whole argument of "the risk of training and they staying" depends entirely on how you gauge risk. In this case, if you have positive evidence that they are unhappy, then you know that the risk of leaving higher than staying.