More seriously, though, budgeting models benefit from the ability to express a random variable at each input so that you can graph the possible range of outcomes.

Yeah this is where most budgeting tools fail me. These tools often provide super detailed categories that the user just has to make educated guesses about- and the onus is then on the user to abide by their guesswork. Not to mention how these tools pretend that expenses all happen in neat monthly/yearly cycles.

Budgets are useful for savings goals, but it's difficult to plan day-to-day spending beyond very high-level categories.

I personally find retrospective diagrams more useful - a breakdown of what my actual expenses were so that I can update my mental model and find subscriptions (or habits) to eliminate.

The way we've (my wife and I) have been budgeting for years is to have a set of fixed monthly expenses, mostly recurring but some one time, and then have a pot of money for variable daily expenses. We then divide that into the number of days of the month and think of this as our "daily allowance" and we try to stay under it to stay within our monthly budget. Some days are over, some are under but if you recalculate it every day it is an easy way to stay on track with the bonus that sometimes you have a lot of money you can spend at the end of the month. We still categorize the expenses but it is more for looking backwards.

I've thought of making an app using this methodology but personal finance apps are like project management apps - everybody has their own opinion on how they should work. I do this with a Google Doc spreadsheet per year with tabs per month which is not ideal but works well enough.