Do you want the surface of the pathways to be relatively solid, as the shingles are, or do you just want to make it dry and separated from the beds? If the former, then you might have to resign yourself to dealing with the shingles, or else see what other materials you can find that would do much the same thing, with maybe a larger surface area, so you only have to pick up dozens, instead of hundreds. I can think of planks, stepping stones, bricks (relatively permanent), discarded carpeting used upside down, or maybe one of the lumber alternatives, Trex, etc., or Hardi plank/panel. There are probably others, too.

If you don't want a hard surface, then you could use fairly thickly laid down wood mulch, sawdust/shavings, straw, pine straw, newspapers layered and wetted with mulch, etc. over it, or landscape fabric, with or without mulch. Most of these would need annual or every-few-years renewal, but you could scrape up the base to add into the beds, as the breakdown result of most of them is compost, albeit not necessarily balanced compost.

I recently visited the community garden at Cedar Grove, Anathoth. In winter, they sow cover crop of vetch/rye/red clover, and in spring just scythe it down and the nitrogen-rich annuals melt without any tilling at all. Similarly, they use a cover between rows in summer, but I can't remember what it was. Maybe annual sweet clover? Anyway, here are a couple of sites:
http://www.cals.ncsu.edu/sustainable/peet/cover/l_mulch.html

http://www.cals.ncsu.edu/sustainable/peet/cover/sum_ann.html

I thought that it was the morter that leached lime not the brick. Maybe I was mistaken.

Cinder block is made with a mortar/cement mix, with sand or the cinders from, I think, coal burning plants, so there might be some leaching from it - I don't know one way or the other. And there might be other deletorious minerals leaching out too. I would imagine that this has somewhere been researched - try googling for cinder block leaching and see what comes up.