One thing you might want to do to increase the depth of your garden is to buy in (or scrounge, if possible) a bale or so of lucerne/alfalfa hay. Split the bale into pads and place them on top of the newspaper before you add the other goodies - with the topsoil topmost.

I think I'd agree with you that the depth isn't enough for the bulbs - and will become less over the winter as the materials begin to break down.

Are you able to plant directly into your soil? Or is it too stiff/poor to work easily? IMO, If you can plant directly do so, and mulch well but not add paper or cardboard as a cover, which could reduce the amount of water getting through. Next season, once they've lost their leaves, you could add lasagne over the top, for bulbs that appreciate being well-fed, and for growing your summer display.

As many of the spring bulbs benefit from being lifted every few years those plantings may be better in a separate bed, or containers. It seems a pity to so deeply disturb a well-made bed just to excavate the bulbs.