If there are multiple services which provide national weather, the ones which offer the best service at the best price will succeed - if they can stay in business. The point is that they must be profitable, or at the very least break even. If people knew the precise cost of the weather service, they could make their own choice about whether or not it is worth funding.
Government is like a conglomerate and each area of interest is like a subsidiary. The problem is, every single one of their subsidiaries is constantly in the red, year on year. They're completely bankrupt! The only way they survive is by lending more money and collecting more taxes, instead of operating like for-profit businesses which create wealth, rather than take it away.
You can still have community funded enterprises, but they need to be affordable, in the sense that there is not an infinite pot of money they can keep pulling from if they overspend. A fire-service, for example, could charge residents of an area a small monthly fee for operating in that area - and those not willing to pay the fee would simply be deprioritized in the event of any fire on their property. Business owners could pay a premium to have a higher priority. Ideally the fire service will operate in profit, and such profits can be used to improve the service or even partially reimburse the creditors if they've 'overpaid' for the service.
This kind of system obviously can't operate together with the public service if taxes are forcibly collected for the latter. People aren't going to pay for something they've already paid for. To put these systems in place, the existing public services need to be privatized (joint-stock where the users of the service can be part owners), and their tax needs to be reduced by at least the amount they were previously paying which contributed to the former loss-making enterprise.
The overall point is that this system can be self-correcting on a per-service basis. You don't need to wait 4 years to pick between two companies to perform all the services if they're doing badly. If a service is doing badly, and there is a demand for something better, somebody will fill that gap. The mere presence of a big monolithic government is a hinderance to innovation in any service that it provides. The only way the services get improved are from the inside - there is no outside pressure to innovate - much less so if it means that the people inside might be risking their own jobs by making the service more efficient.