How do you decide on a startup's business model?

I would focus on the problem you solve more than the "business model" per se. Unless you're talking specifically about how to make money.

The problem with most startups is that they don't solve an actual specific problem. It's a mobile app for example that does something, but doesn't really address a serious need. It just does something that isn't already being done.

If you add some more color to your question I'd be happy to add some more color to the answer ;)

You may need to provide further details, but there are models out there that could help clarify your thinking.
Try the book 'business model generation' to help with this. The dashboard is a tool I use with my clients and we have had really great results. The last client I worked with ended up finding an opportunity they had been blind to and did over a million dollars in the following six months.

It should be obvious based on the vertical market you are entering. A wise man once told me (when people were getting crazy about throwing money at Internet start ups back in 2000) "there is no new business. Its all been done before it just takes on new and different forms."
Unless you are a true transformation change agent and have some ultra unique idea that has never been done before look at what the other competitors are doing in your space, pick the most successful one and model your business to a large degree off of that.

You start by figuring out what problem you're solving, what pain you're alleviating, not by what your business model is. The latter will fall out of the former.

Uber solves a "crappy cab" problem, Airbnb solves an "expensive hotel" problem, Apple solves a "complicated computer" problem. After you've figured out what problem you're solving you then figure out how much you can charge for that solution.

There are various kinds of business models and it is necessary that one chooses the right kind for their company. On a simple note, they should make a value-based judgment on basis of:
1. What is the engageable market share,
2. In what time,
3. At what cost and
4. At what margins as a high-level framework.
So, these would be the four most important questions to answers and must be in-built into the business model.
While building the business model entrepreneur should start with market research and data analysis to have a better understanding of the Market dynamics. Also, one needs to become a Data geek to draw out conclusions to depend on. These conclusions along with analysis on external environment for the competitive landscape, Economic powers & delusions surrounding the sectors and drivers that will structure the future course of Market will give a fair idea of where in the market combinatorics, the focus should be.
One way of increasing margins is reducing Costs, but it could be done only to an extent. Much viable way to increase profits is having Scalability as part of the strategy. Though implementation of scalability should be taken up only after Margins become intricate to the processes of the company, it should also be a part of Vision of the company from initial days. Without having Scalability as one of the factors while designing the Business Model, the start-up will reduce its chances of growing and/or be expanding.
While it is imperative that the entrepreneur has to include factors and metrics to draw out a strategy, it is also important to fore look into implementation part of the journey. While many feels that ultimately everything must be executed, they do have to understand every execution is just one choice out of many. So, it is advisable to have a strategy which includes implementation as a part of it. But be careful to have more than one choice available and be flexible enough to pivot when needed, without compromising on the goals.
As an ending note, a business model is something that will evolve, but one needs to ensure that your business model has the right mix of elements like being repeatable, scalable, predictable and ultimately profitable, thereby creating the value for the business.
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath