This seems like a vague sense of displaced blame, however; so I can understand how people will view it as a red flag: There's nothing preventing her from leaving of her own volition but it's inferred that she has no possibility of choice (without him having to have made a choice) - even though he's already signaled his desires multiple times before.
In that sense, it is an ultimatum but one that tries to absolve guilt by making the "decision-maker" the "owner" of the result/consequence - even though, in reality, both would suffer the consequences of the choice.
Further still, is the impression inferred that she's a prisoner because she's "bound" to him because he needs to "let her go" if he won't fulfill her desire[s]; inferring that, somehow, he - alone - wields power over her self-agency.
That doesn't quite make sense - when a healthy relationship should be two independent people sharing in the relationship "power" dynamic. In other words, it should not be so skewed that one has power/control over life, destiny, etc. of another.
In this perspective, it does seem to be a very troubling red flag.
Also, i do not understand why ever speak about guilt in this case - if there is no commitment then there is no guilt when relationship is dissolved.
If you reverse it - the man says "If you don't marry me, I'll get another dog" - it looks rather less convincing.
Frankly, I find this to be far more a "My way or the highway" kind of attitude than the quoted bit from The Fine Article — you don't seem even to concede the possibility of marriage being more than a "contract", when, for most people, that's among the least of what it's about.
That's not an ultimatum, though I can see how it might read that way. She felt a need, which he wasn't meeting. She told him that if he wouldn't meet it, he needed to let her go seek its fulfillment elsewhere.