Well, as far as the budget of Trek goes, I believe the "slashed" budget of the third season is one of those myths that doesn't hold up under scrutiny. If I recall correctly, the original budget for the first season was around $200,000 per episode, or about $1.3 million today. The budget for the third season was $185,000 per episode. Smaller, but hardly slashed. I wish I could find the site where I found these figures. If someone else has a source, that would be great!
While cast probably took a larger chunk of that, keep in mind they didn't have to spend that entire amount on each episode. So cheaper bottle shows like "Day of the Dove" could be used to save money for larger episodes.
I recently watched season 1-3 on Netflix, and the thing that struck me, aside from the occasional 2005 era CGI, is that season 3 wasn't all that bad! Yes, there were very poorly written episodes like "Spock's Brain" and "This Way to Eden", but as a whole it was only slightly lesser in quality than the previous two. It was clear that writers had been shuffled, and Roddenberry took more of a hands off role in that characters behaved uncharacteristic at times, but it certainly wasn't cheap looking compared to the previous seasons.