Answer: Roses are generally grafted onto a vigorous rootstock. From your description I'd say suckers were allowed to grow from the rootstock and some of them grew large enough to compete with the grafted climber. The rootmass that remained after you removed the rose is sending out shoots, and their main purpose is to develop leaves to provide energy to the roots. If you keep the shoots cut down the roots will eventually run out of stored energy and the rootmass will die out.
An alternative to cutting the shoots down the minute they appear is to re-dig the bed as deeply as necessary to remove all of the rose roots.
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