But I've also found that my "feeling" of a session doesn't necessarily correlate with the performance. I wouldn't be surprised for other benefits to have a similar correlation.
However, as others have commented elsewhere, I think the issue for most people is not exercising at all, so for those cases the feeling you get while exercising absolutely matters. I'd say it's probably the most important thing, since if you feel like not exercising, you probably won't if it's not already ingrained in your routine.
And agreed on your second point too, I bike because I enjoy biking and it makes me feel better both physically and mentally. That I get good exercise out of it is a secondary benefit.