I rely on mulch to keep my fruit cage weed free. On its own, it doesn't totally eliminate weed problems, but careful use of cardboard and other mulching materials will reduce the work involved in keeping your fruit bushes weed free by a huge amount.
However I'm not sure why you couldn't use your mulching strategy without taking everything back to ground level. As others have explained, the bushes will rejuvenate but you will get one season without any fruit and a second less productive year. You will probably be back to full or near full production by year three.
My suggestion would be to give each bush a severe pruning but not going as far as ground level. Take out every branch that is old, diseased, or overcrowded and take out a lot of the young growth too, especially if thin and weak. Aim to leave just four or five healthy strong young stems that are well spread out. After this the bushes will be comparable to what you would get after one years regrowth from ground level.
You should have enough room to be able to embark on your weed removal and mulching program but will save yourself a full season in getting back to full production. There is also an opportunity should you choose to leave just two or three stems on some bushes (say your redcurrants) and train them into cordons rather than stooled bushes.
Whatever you choose it is a great project. I hope it goes well.
Cheers