I think there is merit to OP's idea. Given that reporting is a completely thankless task, and we rely on it to keep the forum semi-readable, we should be making reporting as easy as possible for the few users who are consistently doing it. I wouldn't remove the cooldown altogether, but certainly linking it to good reports rather than to activity might be a good idea.
I just think tying the cooldown to activity is the wrong way to go about it and it should at least be tied to something related like number of good reports. Activity does not mean a member of the forum is going to report correctly where as if it was connected to number of good reports we already know that this member has been reporting correctly. If someone abuses it after reporting 500 good reports then they can be dealt with by the staff but at least we had 500 good reports from them before they started abusing.
Activity is something which is earned by just being around and posting where reporting takes time and effort and as you said is a thankless task. The forum should be encouraging people to report and not putting restrictions on those who want to help. 4 seconds sounds a lot more tolerable but I can see myself hitting that too. I don't see why the cooldown cant be completely removed for those that report accurately and regularly.
While we're at it, please add "Report to moderator" to user post history just like we have it on the Patrol page. That would speed up reporting of certain types of shitposts, e.g. when user post "good project" or spams some shitty service all over altcoin threads.
This would save a lot of time too as there are a lot of members who just keep on posting the same rubbish and sifting through their post history would be tiresome.