Anyway when you say "Furthermore, it's not only about whether 0.05 is the sole criteria, but also about whether it's a useful criteria for us to highlight at all, depending on whether the anchoring effect of it is damaging relative to alternatives." I love this analogy of the null hypothesis vs. alternative hypotheses in frequentist statistics to the 'anchoring effect' cognitive bias that you try to work to your advantage in marketing and sales and negotiation or management. https://en.wikipedia.org/wiki/Anchoring_(cognitive_bias)
If you don't want to be tied to p-values and you only care about downstream decisions rather than quantifying beliefs, you can use some ideas in decision theory https://en.wikipedia.org/wiki/Decision_theory
For example, maybe you are deciding between two alternative ways of doing something. You don't know which one is better, and you are confronted with not only the decision of which one to use, but also with the decision of whether to experiment (with A/B testing for example) to be more sure of which one is right, versus whether to exploit the one that you currently think is better. This is the multi-arm bandit problem and it doesn't necessarily use p-values so your intuition is right! https://en.wikipedia.org/wiki/Multi-armed_bandit
Maybe that's not your situation. Maybe your situation is that you have an existing business process and you want to know whether to switch to one that might be better. Someone might say it's a p-value problem, but again I agree with your intuition that it really isn't best to think about it that way, especially when there is a cost to switching. Instead, it's a more complicated decision that depends on what is the switching cost, how much better you think the new process would be (including uncertainty of it), and what kind of business horizon you care about. There might even be a multi-armed bandit effect again even in this situation, where you also have to weigh the costs of reducing your uncertainty of the switching improvement or even of reducing the uncertainty of the switching cost itself.
Anyway, these problems do involve concepts from probability and statistics but it's for sure true that the decisions don't always reduce to P < 0.05 at the end! Good luck best wishes living your best techbro life!