First, your math and some arguments are off. The price is not arbitrary high as it still depends on demand. Its not like they could charge billion dollars per ticket if maximum willingness to pay of any consumer is \$10. Next, the math is of because government in case you are talking about would have to pay \$1 per ticket, meaning government subsidy would have to be \$9. 

In your case there is no way how to identify who are the marginal consumers. If there were already the theater itself would just price discriminate and charge lower price to those customers. If this is possible theater maximizes profit by charging first 5 people price 2 and next four 1 and earn \$14 profit, and also the last ticket could be sold for whatever the next person is willing to pay, which would also maximize total surplus of society (which is CS+PS).  

So for your scheme to work government would have to subsidize each ticket by \$1 and total cost of subsidy would be \$1 x 9= \$9.

What even more government is not just an endless well of resources one can tap to. Government gets money through taxation or borrowing or seigniorage (i.e. $G=T+B+\theta$). I will focus only on taxes, because otherwise this answer would get too complicated, the answer loosely actually applies to all three cases although mechanism in each case are different. 

In order, to pay out that subsidy government would have to levy some tax and most taxes are distortionary. This means that tax usually causes greater costs to the economy than the value collected. So to pay for \$9 subsidy government first has to take away \$9 dollars of surplus from some market from someone, and in addition usually, save special cases, there will be additional \$X cost (the amount of X depends on market conditions). Borrowing is just intertemporal reshuffling of taxes, and seignories is hidden tax on anyone with money, they both generally lead to some sort of deadweight loss at some particular markets. 

So you are getting \$7 extra surplus by paying \$9+\$X surplus. 

This being said the subsidy can be justified on normative grounds. For example, if you take away \$9 + \X surplus from coffee market, to subsidize theater market, because you believe theater market participants are so much deserving of that \$7 surplus it justifies taking \$9 of surplus from people engaged in coffee market then the policy makes sense. But it is typically negative sum game.  


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> This is a solution for the underutilization of capital that we see in the free market, empty seats on buses and trains, empty roads, empty theatres, empty stadiums, games unplayed, films unwatched, online courses unlearnt from, medicinal and technological innovations unused, etc.

This observation is actually not completely correct. These are not necessarily underutilized because of fixed capital cost.

- Fixed capital costs are sunk ex post. Over long-run firm has to recoup them but they do not necessarily determine intratemporal short-run decisions. For example, game development has high fixed costs but very low marginal costs, so usually you see high price when the game is new and people with highest willingness to pay don't have the game, but over time the price goes down to be closer in line with marginal costs. 

- Some seats are empty just because it is impossible to predict demand. Demand can only be either estimated or guessed in advance. So you can't simply in real life say that when price \$2 per ticket exactly 5 people show up. Depending on uncertainty perhaps just 3 people would show up or 8. 

     This is for example why in areas where it is very important to fill each seat, such as on airplanes, overbooking occurs. If demand would be certain there would be never a need for such practice. 

- Some empty seats are actually direct result of government subsidies. Most governments around the world subsidize public transport or subsidize culture etc. This is usually done due to some normative reasons such as support of art that most people don't appreciate, or equity reasons to give transport options to some smaller and unimportant towns. Such subsidies are typically paid per route or theatre etc. This often result nearly empty busses or trains on some routes, or nearly empty plays of contemporary art show.

So you can't just immediately jump to conclusion just because you anecdotally see empty bus or theater.