Does a spire's starting upgrades count against their upgrade capacity?

Yes. A Spire's upgrade capacity lets the player know how many Upgrades a Spire can have under it before it can stop gaining Upgrades using build options. For example, a Dispatch Platform has an upgrade capacity of three and starts with an Attack Upgrade and a Range Upgrade. The Dispatch Platform has two Upgrades already so it can only gain one upgrade using a Build Option. A Blade Wall Spire from the Market has an upgrade capacity of zero and starts with three Attack and one Fortification Upgrade. Since it has an upgrade capacity of zero, the player cannot use a Build Option to give the Spire any new Upgrades. Other game effects that would grant it Upgrades that explicitly say ignoring upgrade capacity can still give this Spire new Upgrades.