First, no one forces you to communicate a single number. There are things like confidence intervals, you know.

Second, misinterpretations are inevitable with any highly complex concept. Bayesian tools, your suggested solution, is even more complex than p-values so I wonder how you'll be able to communicate it without misuse and misinterpretation.

Third, there is nothing magical about 95% or 0.05 and this has been said so many times that my fingers ache when writing this. See #2 above about why the same thing will happen with any other approach you propose.

Fourth, there is no alternative. Bayesian probabilities are not an alternative, and you can read a ton on this. You can also read how p-values fit in a decision-making framework, something you seem to be trying to accomplish with Bayesian probabilities, therefore completely mudding the waters.

P.S. Citing Optimize's resource hub on frequentist statistics is just unacceptably bad. It's the most misinformed writing on the topic I've seen over the past ten years. And I've seen plenty...