Pretty much every developed country I am aware of spends a considerable amount of money every year on supporting their sports teams, theaters, cinemas, museums and countless other forms of entertainment. Usually this is done under the assumption that those industries would fail without the government's support, so funding must continue no matter what.
But are there any developed countries that avoid such expenses and leave it up to the free market to see if those industries can survive or not?
No. There are several reasons for this (some are more applicable than others to specific industry or country):
First, on nation-state and international level, entertainment/sports is a tool of soft power.
Sports brings prestige to the country.
This is especially visible in things like Olympics (where, contrary to the whole Olympic spirit, the main thing anyone ever does is count how many medals any country gets). Sometimes, there's deeper political meaning imbued into this, such as the famous "Miracle on Ice" US/USSR 1980 Olympic hockey game.
This is less of a factor in USA; but most developed countries in the world spend lots on basic sports infrastructure (fields, training centers, sports schools, pokemon gyms).
"Culture" is an important tool to maintain a country's sovereignty - both internationally and internally.
This is very visible in, for example, France. They spend a lot of effort and thus money to support native movie industry, and culture in general. Russia under Putin does this a lot as well (the results are, at best, questionable, but state propaganda always carps how the latest state-sponsored vehicle for corruption ...errr... blockbuster film is "better than Hollywood" - it's not even about culture in itself, it's always about beating Hollywood)
Second, there are purely domestic concerns:
Sports/entertainment is an industry.
As such, it can and does, lobby on its own behalf with the government.
In some cases, the arguments they present are economic ones ("build a stadium here and you will get economic boost and thus raise tax revenue and employment").
This is even more notable with movie industry; where (there was a NPR Money podcast on the topic within last year) states and countries compete in who can lure more studios/production/special effects to their locale via tax breaks/subsidies etc...
Entertainment is popular; and as such represents political power for that interest groups.
As a clear-cut example, witness everyone yelling any time US government thinks about cutting funding to PBS, NPR, NEA etc... - there's a wave of discontent from their consumers who prefer to use Other People's Tax Money to support the arts, instead of their own.
More cynically, people always want "panem et circenses" - bread and enterainment. Since the days of Rome.
And the rulers have always realized that it's much cheaper and easier to placate the populace with "Look at our marvelous national team winning!" than with "Oh, we raised everyone's standard of living 10%".
Part of the underlying reason for this is that, in most of entertainment, it's kind of a barbell situation economically - the very few super winners on high end make most of the profits; but the vast majority is fully economically unsustainable. Without patron support (which used to be wealthy nobles, and now includes The State), most entertainment couldn't survive as economically viable enterprise.
Short answer ： No.
Simple explanation: Developed country simply treat arts and sports as the same category of any pure science project. Policies are enacted to encourage participants in such fields (with education, scholarship, grants), this also involves encouragement to privates sector to contribute their share, e.g. tax incentives, grants, etc.
With healthy developments, some sports/arts are able to grow and stand on its own as a new industry. Never underestimate potential and synergy that brought from such industry. For example, sports equipment is one of the main consumption for industrial composite material. Another extreme example of art application is origami, NASA engineers get the inspiration from origami to fold the satellite solar panel to fit into the rocket payload.
Essentially no.
One can even consider it a requirement for a developed country to support the "entertainment" aspects on some level. Now to be clear they way they support those industries can be very different.
Usually this is done under the assumption that those industries would fail without the government's support, so funding must continue no matter what.
Not at all true. US Football isn't going to fail without their tax incentives and subsidies. They will continue to exist and make money, and that's just an example. They get funding, on many levels for many different reasons but rarely is it because they would fail without it.
A great example is the arts during the Renaissance. Look at Michelangelo’s "David". It was commissioned by the city of Venice, and became a political symbol for the city.
Want something older, the Roman Colosseum was more then just a stadium, it became a symbol of power and [Roman} civilization.
Today it's much the same. Come visit New England our football team is awesome. What is Brazil known for? FIFA Cup wins of course? Sydney, isn't that the place that has that fancy concert hall thingy (Sydney Opera House). It's also where P. Sherman lives (42 Wallaby Way).
These things create prestige and notoriety for their governments and nations. They give a way to "advertise" their country to others and a way to instill values into their culture. That is usually why they are funded on some level, not because they can't make it on their own, but because these things (arts and sports) are important to the people of the country, one way or another.
It's also worth noting that sports, the arts, the sciences etc. are a great "thing to do" as opposed to rioting and revolt. It also helps build community and an idea of "this is us" that is hard to wash away.
Your premises seem wrong.
No country spends a significant amount on culture.
The US Federal government spends less than 10$ per capita on all arts and culture, Canada spends 140$ per capita, France 110 Euro per capita, etc.
All of these are a small percantage of the GDP of the cultural industry in the country, the federal government tax revenues, and of the GDP of the nation in question.
In the USA in particular, cultural industries generate over 2000$ per capita and get federal government subsidies of 10$ per capita. That is a half a percentage point.
By any reasonable measure, the USA has no significant federal spending on culture compared to the industry as a whole.
In other nations, government spending reaches single or low double digit percentages. Often this is due to a national broadcaster whose mandate includes reinforcing national cultural identity (In France, for example, half the culture budget is for national broadcasting)
Government spending of 0.5% (USA) to 7% (France) of an industry's size isn't a "significant amount" of that industry, and next to nobody would assume the industry as a whole "wouldn't survive" without that kind of expenditure.
On the other hand, the specific sub-areas the government supports probably wouldn't survive in their current form without government support. As a concrete example, a national broadcaster with a specific mandate and restrictions would end up looking more like commercial broadcasters, which exist in said nations, without direct government support. There are nations without national TV broadcasters (like the USA) that agree with this prediction.
