If they are pretty grey and woody you might take out the very oldest one in the poorest condition in hopes of getting a new main basal cane to grow. Long term you need to think about renewing the main canes periodically.

I am a little confused re why she only blooms in June - HMF says MGS is a hybrid tea, so I would think she is capable of blooming more.

Pruning her a lot may cause her to grow and possibly bloom more - I would feed her well as well. I have several climbing roses on my house (three of which were up to the three story high roof). We had the house painted, and so I had to cut all of them down to about 6 feet high. We wrapped them in burlap, and my DH threatened the painters and other contractors with death if they hurt them. The painting, and other repairs to the house went on for 5 weeks, and I was hoping my roses would survive. Within 4 weeks they had all put out new growth which burst through the burlap wrappings! After it was all over and we took off the burlap, they all grew amazingly, and within 2 years the largest ones had made it back up three stories to the level of our roof. They put out new basil canes, and some of the smaller ones which had sort of declined over they years were re-invigorated, and now are doing much better. I came to the conclusion that it had really revived all of them. Here is a picture of the banksia lutea 2 years after being cut down to 6 feet high:

April -- May I say -- that is drop-dead gorgeous . . .

Jeri