I interpret your quote to mean that the “Command Word” is for activation. It doesn’t specifically say that you have to use your action to keep moving it, much like you don’t have to command a mount with your action.
There are some spells that specifically require using your action or bonus action to continuously move it each turn, such as animating moving illusions with silent image or major image, or moving spell effects around such as with spiritual weapon or Bigby’s hand. The flying carpet doesn’t specify anything like this (in the same way that those spells do) so there is no reason to think that further actions are needed beyond the initial “command”.
If it was intended to work this way (moving it with an action each turn), it would make the flying carpet kinda useless, especially compared to the spell fly or a flying mount (such as via find greater steed, for example) since you wouldn’t be able to do anything else but fly. As pointed out by @T.J.L., larger carpets with many on it wouldn’t necessarily suffer this problem as only one of them needs to be “driving” with their action.
I’m also reminded of a Beast Master ranger’s animal companion, which can be told to move to different positions via commands which require no action. Of course, the flying carpet doesn’t specify, so that could be considered an argument to claim that it does require an action since it doesn’t say “no action required by you”, but I think this is better taken as setting a precedent for verbal commands not requiring actions, and besides, the flying carpet also doesn’t say that it does require your action.
Also pointed out by @Premier Bromanov:
(T)here is a distinction between “Command Word” and spoken directions. Whereas the Command Word is coupled with an action and spoken directions are not, and are also generic.