To me this is a really interesting generational communication style question.
What I'll admit to criticizing as the Teflon PM approach is a very nuanced skill, but it is about enforcing that boundary instead of establishing a foundation of trust.
It's philosophical whether value comes more from the quality of relationships or from individual leverage, and of course that's dynamic, but this underlying tension between what has been problematized as transactional vs. collaborative communication, and whether it actually creates value or rides it out is the defining cultural question of companies today.
I really like your blog, and I"m a bit chuffed you've responded directly. So, tactically, yes, "how would you prioritize your request?" disarms their urgency, but to provoke, I'd say it doesn't take the necessary risk or build a relationship that yields value.
I'd like to propose a controversial view on this and to understand the tension better.
It should be on members of customer facing teams to demonstrate that they are capable of saying "no," before coming to product with a request. Not every time, or even often, but a sales guy recognizing he's losing the sale and only brings in support from product and elsewhere to diffuse the blame for it is the basic sales anti-pattern.
I get that for a relationships sake they need to use the "let me speak to my (product)manager" routine to deflect frivolous objections from the customer, but unless I know what you were willing to say "no," to, it's difficult to value what you said yes to.
This is why the relationship with salespeople is so critical, as you need to have an empathetic view of that dynamic, but if the sales persons solution to everything is to claim powerlessness and blame others, we need some neutral language to describe that pattern so we don't have to sound it out and re-litigate it every time it happens.
The question, "what did you say no to?" seems like a practical filter.