I just did a serpents run (the T4 scenario that pops) and we rolled over the destro. Only one problem -- everybody was sitting at the destro spawn instead of capping! So I got on my horse and capped the salvaged parts six times or so (I was the only one). I ended up with by far the lowest scores on my team, but the bonus +600 renown and 12000 xp meant that I ended up with far more points than if I hadn't capped.
Purely selfishly, capping objectives gave me more points than if I'd healed the whole time. Now the spread of points was by no means fair because I (the only guy capping) got the fewest points on the team. If I got jealous, I might not cap next time so that NOBODY got the extra points (and to be honest, running back and forth six times isn't my idea of fun).
In short, capping doesn't increase your individual score one bit, but it increases everybody's scores more than simply fighting (even if you win in a landslide). My suggestion is that if everybody ignores the objectives, cap once or twice and then suggest that somebody else take a turn.
It's an interesting setup because psychologically, jealousy is much more powerful than the expectation of a minor incremental reward -- for many people, it's hard to do even what's best for themself if it means everybody comes out even further ahead. I've read a number of studies showing that people would vastly rather be paid $30,000 among coworkers who were paid $25,000 a year than make $50,000 among coworkers that were being paid $100,000 a year. Same effect.
Hopefully, developers will give 50 RP or so for capping to make it lucrative, but of course then people will be fighting over objectives to the detriment of the team. Silly peoples.