I think you are ALL right on certain points.
As long as you are overweight and riding with more fit riders, you will get dropped on the hills. Even if you aren't overweight, larger riders with inherently more body mass are going to have a harder time getting up hills than those little wirey guys that are 167cm with <5% body fat and lots of power.
Your climbing will improve as you drop weight, guaranteed, but you should keep climbing in your workouts (50% or less of your workouts). Exclusively climbing may improve your climbing ability somewhat, but it doesn't allow for recovery time (climbing is hard on the back and the knees). You should also have work outs on the flats to build power (sprints, etc.) and some easy spinning after your hardest days.
As funknuggets suggested, seated climbing is probably the best way to strengthen your core muscles. I don't have a HR monitor, but one climbing work out I have used involves increasing resistance for brief periods on a climb:
1) Start the climb in a low gear, spinning at around 85-95 rpm
2) After 2 min. shift to a higher gear w/o standing up w/ cadence around 50-60 rpm; maintain for 2 minutes
3) Shift down to your starting gear and cadence, assuming a constant grade, and maintain for 2 minutes.
4) repeat 2 and 3 until your climb is finished.
The higher gear-lower rpm phase builds power, and the lower gear-higher rpm helps you focus on your pedal stroke and recover.
I've heard Lance Armstrong uses a similar climbing workout (friend of a friend told me this, so it may be [censored]). But I've used it with improvement evident.
I'm sure there are some other workouts out there that are just as good, but hope this helps.
Regards