Relationships without prescriptions

My friends don't get to veto my romantic partners either, so I don't see how this is any different.
I would say it is different because we are nearly always less commited to our friends than our partners. We rarely organise our lives (move house, find a new job, make long term commitments etc) around friendships, but we commonly do with romantic relationships. So my point was that if friendships can be tried and tested and threatened by other relationships, it must be more true that romantic relationships can be also.

So... what was going on in that friendship that meant that a new romantic interest ended it? Maybe the friend with a new love interest likes to get involved with terrible people to be around socially for the other friend. That's a doomed friendship; these people are incompatible as friends. Maybe the other friend can't handle seeing their friend fall in love. That's also a doomed friendship. So on and so forth.
You seem to be saying that any time that another person catalyses an issue in a relationship, whether romantic or not, that relationship is clearly struggling or even "doomed". Yes, I agree that problems that arise because of other relationships can highlight incompatibilities. But people also change their minds, make bad decisions they later learn from and regret, or just develop and grow as people. If a friendship struggles because of a new relationship, that does not mean the friendship was necessarily "doomed". It could be the end, yes, but it could be a temporary blip in an otherwise happy friendship/relationship. Or a difficult patch that you come through together.

A relationship that isn't flexible enough to accomodate the kind of changes a new romantic interest brings is a relationship that is based on the particular circumstances that exist in the moment. It's not going to be long term, because if a new romantic interest doesn't change things, something else will.
There is no relationship that could survive any change, whether that change comes about via another romantic relationship or not. When you say "the kind of changes a new romantic interest brings" that actually covers a huuuuuge range of possible changes from barely noticeable to seismic. Some changes are just easier to be flexible about than others. No relationship is infinitely flexible.

I'd just emphasise that someone being rude to me doesn't threaten any of my relationships. How my romantic partners deal with someone being rude to me tells me something about my relationship to them.
Yes, your partner's response to that tells you something about them. But that doesn't change the fact that before the situation is dealt with, the problem (and therefore the threat) lies with the other relationship. One doesn't negate the other. The rudeness was the threat, how my partner dealt with it came next.

I don't know why people are so keen to prove that relationships don't affect other relationships. My relationships are not self contained bubbles - my partners' friends, family and their other partners all have an affect on my relationship with them. My whole life affects my husband, and that includes my relationship with my boyfriend. Therefore, my relationship with one man is affected by my relationship with the other.

What if we turn it around and look at the positives instead of the negatives - the boons rather than the threats? My marriage has been better since meeting my boyfriend, and we love each other more. My relationship with him makes my husband happy, and that, in turn, has strengthened the bonds between all three of us. If outside forces, relationships and people can make my marriage better, then it stands to reason that they could also make it worse.

Are you saying that there exists some category of people whose relationships could never be threatening, and that there is some way to identify these people?
I'm not sure I can answer that. It's a bit like asking "is there some category of people who can never be unpleasant, and is there some way of identifying those people?" Some people cause problems and some don't. I go down the path of thinking the best of people until they give me a reason not to. So I consider no one and no relationships to be threatening until I see evidence that I am wrong.