It does appear to be sending out all new growth higher up on the plant instead of at ground level. If it were mine I would take cuttings from new growth(after flowering) and aim for at least 1 bud eye above cut but planted underground when rooting.I'm not sure how tall the plant is now but perhaps you could replant it deeper with the section sending out the growth at ground level (wait until you get some rooted cuttings in case you lose the mother plant) I have the same rose known as Thornbury Castle to me , it is lovely and quite easy to root.

The length of the bare stem from ground level to where the new stems are growing is 12in/30cms, so it would have to be re-planted a foot deeper in order to 'loose' the main stem.

Your idea of taking further cuttings is a good one but, with us, they take so long to produce a decent size.

My other concern is that the stem probably can't support much more top growth.

That shaft looks pretty sturdy to me. I've often gotten grafted roses that looked like that and they do just fine. The only other thing I could suggest would be to dig it up and bury it deeper and see if it send up some new basal breaks then.

I take cuttings with only 2 nodes. The bottom node leaf is removed, and the top leaf is retained. The root itself will form new buds, and you should get canes coming up from the root. See the object16 thread about cuttings taken in the fall - his cuttings are all taken with two nodes only, and I have seen the bushes lately, and they are sending up immense numbers of canes from the root.