Growing Things: How to manage with suckering roses

Q: I have a Therese Bugnet rose that was planted in 2017.  The main plant appears to be at least 90% dead, however, there are about 6 little bushes around the main one.  I think that I would like to remove the original big bush and keep the babies.  Should I move them or leave them where they are? Would they actually be attached to the original plant?  Is this normal for roses?  None of my other roses have done anything like it before.  Also, I have a Polar Star ros