Does any of this address the issue of why people aren’t posting a budget? I often see what looks like fear of being overcharged, e.g. to the limit of the potential budget.
A budget is an early commitment whereas most of us would rather be given an estimated price we can then decide if it is acceptable or not because that’s the point when we actually try to get the budget.
So I would rather the term value, or something similar was used. It is less committed and more generic than budget. It more clearly reflects the negotiation between the two parties about what the work is worth, i.e. buyer says “It’s only worth that to me” and seller says “I won’t even get out of bed for less than $x.”