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Usually there are two or three 6 or 7-handed $5-$10 hold'em games. Rather than break one table, they argue "There aren't enough seats to break the third table, a player may have to sit out."

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This isn't a trick. This is fairly common. One thing you don't mention is that when this happens everybody wants the other table to be broken, no one wants there table to be broken. 7 handed isn't all that short anyway, but if one table just refused to play they would break it. Depending on the time of day and whether they expect more players it might not make any sense to break the games. If you have 3 six handed games going during the dinner hour and you expect to have a bunch of people show up after dinner you don't want to break any games because they will soon be filled up or near filled. Of course if its 3 AM and you don't expect any new players to show up for hours things are a little different.

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For example, if the table is reduced to 5 players and you are doubling the number of hands you play per hour, you end up paying twice as much money to the house, on average, as you would in a full table. This equates to a $12 max rake in a full ten handed game.

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Do they take $6 rake per hand or $6 max? Your statement would only be true if they are taking $6 hand. Certainly a short handed game gets more hands out per hour but many of those hands will have small pots that don't generate max rake (unless they take $6 regardless of pot size).

OF course that rake is way to high to play anyway