I absolutely reject the idea that the series had a substantial decline. And, based on your summary of "Kirk gets captured and screws a girl every week," I question whether you're even watching the series.
I agree with you Vger23 that
Kirk gets captured and screws a girl every week may be an over-simplification, but even as an unbiased uninformed child who knew nothing about 1st 2nd or 3rd seasons, I could tell the difference right away when an episode came on.
Generally speaking, Spock seemed completely bored, flat and mono toned. McCoy appeared to be mostly grumpy all of the time (less "fun McCoy" like in
Shore Leave or thoughtful/wise McCoy in
Balance of Terror) I also noticed this in the Abrams films. Urban (a good actor) was forced (through no fault of his own) to be mostly the
"Dammit Jim" grumpy McCoy. For me, the McCoy character always had so many more layers than that. I suspect Urban thinks so too, but as the old saying goes
"if it ain't on the page, it ain't on the stage"
Kirk just lost his edge for me. In Season 1 he had a look in his eyes (that Shatner himself talked about in an interview) I can't remember what the Shat called
the look but we'll call it the
Eye of the Tiger for the sake of argument.

You saw that look a lot in Season one. You truly believed this man could command a starship.
In Season 3 that look seemed to be missing. It's a look that Chris Pine (a good actor) never captured for me.
In Season 3 Shatner just wasn't as fit, and much less dynamic than he was in Seasons 1 and 2. Maybe he was simlpy bored and less hungry as an actor?
Scotty became a caricature of himself, In Season 1 he was
mostly cool and confident and focused on engines. In Season 2 he actually commanded the ship, and very well I might add. In season 3 he was almost always the panicky
"Captain what do we do?!!!!" Scotty.
Even the music seemed a bit more bland to me. (I know Alexander Courage quit after that royalties incident where Roddenberry wrote non existent lyrics to Courage's theme song which took royalties away from Courage) Perhaps that had an effect. Dynamic scores like
The Corbomite Maneuver, The Naked Time, Balance of Terror, Charlie X, Who Mourns for Adonais, Amok Time, and The Doomsday Machine just werent there in Season 3 for me. There were definitely a few bright spots in Season 3, but not nearly as many as in the previous seasons.
The cinematography in Season 3 was MUCH LESS inventive than it was Season 1, which had episodes that actually looked like feature length films of the time, whereas the 3rd Season used a a lot of medium shots and flatter, uninspired lighting. Clearly this was due to the loss of Jerry Finnerman. According to memory Alpha,
"Finnerman remained with the series until early in the third season, quitting after production of the episode "The Empath". He left partly because he felt the series had become ridiculous"
I think that sums it up nicely for me.
Actually, a lot of members here have made some excellent points I hadn't considered. Its remarkable how a show could decline so much in so short a time. The only other example would be Space 1999 Season 1 compared to Season 2. The Frieburger touch perhaps.
In the end I think it demonstrates beautifully, how much of an effect the loss of several key people, can have on any project.

Spockboy