Let's unpack the discussion into two parts: how we think of free market and business models.
Free Market Economy
The question is pretty clear in drawing direct comparison between free market economy and centrally planned economy. The problem is that this comparison is very difficult to observe empirically, either because
- an economy that's completely planned has fewer business models for any given industry almost by definition, since each model is designed from scratch by the planner, or
- there are very few economies that are centrally planned entirely; comparing the rest of the world to a handful of countries means you are going to observe less variety in the latter.
A nice historical (some might call anecdotal) example that follows the comparison you drew would be the story of the two Carl Zisses following the split of Germany. The YouTube channel Asianometry has a great explainer on this story; you could also check Shapiro (1973) for an academic exposition. This is a country-to-country, industry-to-industry comparison down to the company level that has less of an apples-to-oranges problem, however it's just one data point, and as Asianometry's explanations shows, the divergence of the two Carl Zeisses is driven by politics as much as it is by economics.
A more fruitful approach to the question might be to drop the dichotomy of free market versus central planning. The forces of a free market can be found among entrepreneurial street vendors in Cuba and Venezuela, and the power wielded by monopolists or oligopolists in major industries in the US are not entirely dissimilar to those enjoyed by central planners, in particular because such oligopolistic powers also have their political roots. Faccio and Zingales (2021) study the telecom industry and find that companies enjoy higher prices from regulations that are less pro-competition in countries where they have more political connections, such as the US. The effect of such market power with political backing can be felt directly by consumers; the Chicago Booth Review highlights one number from their study:
American consumers would gain \$44 billion–\$65 billion if the US telecoms market were as competitive as those in Denmark and Germany.
It might be surprising for people who don't follow this line of research or who haven't lived in both the US and somewhere else, that the US no longer leads the world in terms of how free consumer markets are. Gutiérrez and Philippon (2018) document a series of policies in the US and Europe to explore exactly what led to lower regulatory barriers in the Europe than in the US.
I don't have a clear definition of business models. In the strictest sense, this might mean how a company produces its output (goods or services) in a given industry, but it could extend to what industry or product space a company chooses to enter before even makign decisions on production. If we stick to the strict version of the definition, than I would actually conjecture the opposite of what you proposed: free market, defined as having greater competition (instead of being the opposite of central planning, as discussed above), would imply less variety in how a given product is produced, because competition weeds out all the less efficient production methods and retains the one with best profits. An inefficient business model is more likely to survive with political ties or even simply by inertia where there's little competition, such as high prices for phone plans in the US or the monopoly of a local mom-and-pop shop.
The reasoning would also depend on the industry being discussed. Comparing street vendors in the US and in North Korea would be very different from comparisons in the tech industry, which is probably not accomplishing much in North Korea except for maybe military applications. Observations like the lack of a vibrant tech industry in North Korea might be what led to your observation that free markets are better for variety; but these countries, aside from being centrally planned, are also incredibly poor. It's hard to say whether the boring tech landscape is the product of poverty or central planning, or maybe the poverty itself is the result of planning. So your reasoning might be correct in a broader sense: central planning is bad for growth and thus people have less stuff to choose from.
If we are going to keep the discussion of business models to how a given product is produced, across varying levels of market competition, then results really depend on the industry we are talking about. Take a specific example from the tech industry: the fiercely competitive market for the hard disk drive. The products form different companies in this market are mostly similar compared to, e.g. varieties in social network apps, so the comparison makes more sense. Do we observe less innovation when the government is involved in consolidation decisions? Igami and Uetake (2020) find "plateau-shaped equilibrium relationships between competition and innovation", which means the degree of innovation doesn't vary a lot across levels of competition in plausible ranges of market conditions. So at least for this large and important market, we wouldn't see much variation in business models across sensible levels of government interventions.
- Faccio, Mara, and Luigi Zingales. "Political Determinants of
Competition in the Mobile Telecommunication Industry." The Review of
Financial Studies (2021).
- Gutiérrez, Germán, and Thomas Philippon. "How European markets became
free: A study of institutional drift." No. w24700. National Bureau of
Economic Research, 2018.
- Igami, Mitsuru, and Kosuke Uetake. "Mergers, innovation, and
entry-exit dynamics: Consolidation of the hard disk drive industry,
1996–2016." The Review of Economic Studies 87.6 (2020): 2672-2702.
- Shapiro, Isaac. "Zeiss v. Zeiss—The Cold War in a Microcosm." The
International Lawyer (1973): 235-251.