I trained as a chartered management accountant, and budgetary control was a major tool in controlling business finance dating back to the founding of my institute in the 1920s. So, one hundred years later, how is it doing?  Depends on size. As a management tool it becomes more relevant as a business gets bigger. Most businesses by number are one-man or woman outfits. The only time to create a budget is when it is demanded by a bank before granting them a loan. Banks believe this will force the customer to think carefully about cash flow. However, once the loan is obtained (or rejected) the budget is never referred to again. Using it to control the business is quite rightly an alien concept. The main management tool here is the bank balance. Cash is king.   Organisations with say 200 or more employees are more complex and almost always have a system of budgetary control together with monthly management accounts. They can’t be managed blind. Refer to the management guru Peter Drucker’s most famous quote, “You can’t manage what you don’t measure”. 
Cont:
Businesses sized between these two extremes are a mixed bag. Budgets are a base line from which to highlight deviations from planned income and expenses. How the business is structured, and the professionalism of the Directors determines its effectiveness in maximising profits. For example, a hospitality chain will have many managers, so the Directors need visibility on how the performance of each location compares. Another example is a business whose owners have non-executive roles and perhaps manage by holding a monthly or quarterly board meeting. In this case, numbers tend to be meaningless unless compared against a previously approved plan.  Another use of budgetary control is for governance purposes. My firm acts for several charities and they all use budgeting to report both to project funders and to the charities’ Trustees. Income, in most cases, must match spending. (Kwasi Kwarteng uses a different system).