How to properly clean raw chicken cut in pieces? In the country where I live, whole chicken is the cheapest protein you can buy and I cook it every week at home. I always ask the butcher to cut it into pieces for me and I use it for stews, grilled, fried, cooked in the oven, anyway you can imagine. It is cheap and incredibly versatile, selling for around 1,6€/kg. The closest meat product would be pork chops selling for around 3,5€/kg.
Because the chicken is cut in pieces it comes with small bits of fat and chicken insides that don't taste or look good in my dishes. I remove these with a knife and then wash chicken under running water to remove the smaller bits. If I don't wash the chicken I get smaller bits of fat and blood vessels in my food. But I was horrified to stumble into articles that say washing raw chicken isn't safe. It spreads bacteria into the countertops and the sink and there's evidence that the bacteria can remain after the kitchen is washed. Everyone agrees washing raw chicken isn't safe. The alternative is patting the chicken with kitchen paper, but this will make it hard to remove the small bits of blood and whatnot that are holding on to the chicken. What can I do to safely clean my chicken?
There are several studies published on the presence of campylobacter (which is the bacteria of concern) in chicken. You didn't specify where you are, but the findings are fairly consistent worldwide.  Chicken has a much higher incidence than other poultry products, and other proteins, though campylobacter can be present there too.  Of course, this is all made safe when cooked.  So, the concern with washing is cross-contamination, as you correctly point out.
I would use a paper or cloth towel to remove any bone fragments that might be dangerous if chewed or swallowed. Then dispose of the paper towel or wash the cloth thoroughly. Bits of fat can be removed this way as well, or cut off with a knife.
As for any remaining blood or parts that are not easily wiped away, just leave them.  Try it.  See if you think they make a difference in the final product.  I assume that you will not even notice them... or you can scrape that away after cooking.  If that is not possible for you, perhaps search out another butcher who will prepare it the way you like... or give your current a butcher another try, but this time ask him or her to remove the parts you don't like.
This all might be a bit of extra work, but you will soon find a rhythm... and, for the money you are saving, it might be well worth it.