I think a better comparison for Captain is Prince. We side step the whole issue about how much is it worth to interact with $5s and up (which is obviously the biggest value with Kc compared to Cap), they both give you benefit of playing the card at the turn start (e.g. can draw before Silos), and they both allow you to gain "unique" abilities on boards (e.g. generating >5 card hands with no net draw).
Prince requires you to buy and collide the $4. Prince is turn delayed from when you play it; and worse is a always a dead terminal when you play it. Prince cannot use durations as durations (e.g. Prince of Churches or Gears can work, but has to give up a lot of their utility). Prince is locked into what you put on the mat (e.g. once a village, always a village). Prince costs $8.
What can, and usually does, make Prince better? Prince never gets destroyed by piles emptying. Prince gets the amped version of the card twice as often. Prince never bottom decks once in play. Prince can play multiple terminal on its own.
Cap can introduce a huge weakness into your deck. If using him with cantrips is your only +action, the other guy may be able to tank your deck just by piling the cantrips, which are often cards you want to build up to $6. It is worse if you are using him for your draw. A functional engine vs a tanked one is something like 8V per turn; avoiding that will often be worth double.
Okay, but say you have a board where there are plenty of good Cap/Prince targets, how much is Prince better here? Well for $12 with Cap I get one "normal" version of the card and one free +1 action/+1 card effect. With a $12 Prince setup I get one $4 and one free +1 a/+ 1 card effect. Double Cap is then basically gaining the advantage of having one more $4 in deck. Of course this also means you need to draw your "$4". Of course getting two Caps will often completely offset Princes of penalty of coming online a turn later. Advantage to Cap, but not by all that much. Going to $7, though would likely make me prefer Prince on most boards; after all on most engine boards the majority of my non-VP gains are going to come on >=$8 coin hands or <$6 hands; not to mention that it is vastly easier to sneak in some sort of "gain a $4" than "gain a $6". $14 vs $12 (or $11) is starting to delay my engine by a turn and that is costly.
Bottom decking is utterly huge. Prince of Smithies is an 8 card hand for the rest of the game. Cap of Smithies needs 12.5% of your deck to be Caps. For larger decks, that starts becoming hard to ensure (e.g. needing 3 for 20 cards) and if you clump at deck bottom you may have a lot of really subpar hands in this shuffle and the next. Like Wharf or Fishing village, Cap is going to be more reliable when your deck starts to green than regular villages and draw … but when it tanks it tanks hard. At high skill play, you should expect at least a few turns with busted shuffles so paying more for Cap than Prince becomes challenging.
Cap, at best, can manage two terminals without support. This is not enough for a Hop deck. It also means you have trouble managing combos. For instance Bish/Terminal gainer can allow you load up on Silvers and then build out to a very high VP/turn Golden deck (in theory 15 VP/turn but easily 7 VP/turn). On such a pure terminal board these are incredibly strong (e.g. Militia/Masq).
Cap at $6 makes double Cap pretty close to Prince of whatever at "$12". Yeah I will buy Cap often instead of Prince on a lot of boards. More often I will buy Cap and then drop a Prince of whatever might run out and cruise. But fundamentally they are around the same functionality and around the same price.
After all, what are the major effect of increasing price from $6 to $7? Yeah it can make early gain more swingy. But more often it delays the gain a shuffle. It also makes it far harder to build out in the standard $16 engine. Getting to $9 - where you can now buy a Cap and a silver, means drawing the Silvers and a buy. Getting to $10 is likely a full gain later (call it half a turn). Getting to $12 is two golds and three silvers. Getting to $14 means gaining another gold and at least half a turn later. In the late game going to $7 means that you will much more rarely have to choose between a duchy and another Cap. Going to $6 means there are a lot more turns where you are nearing the end of the shuffle and being able to increase your odds of a big turn are worth more than locking in 3 VP this turn.
I would need a very good argument to say that Cap should be $7 when Prince/$4 parallels so nicely with double Cap. I cannot say that double Cap or Prince is generally better, but I would be surprised if I didn't buy a lot fewer Caps/more Princes if Cap went to $7.