Could Calibre (or any other software) generate an OPDS feed? I know that at least KOReader has support for OPDS[0], maybe vanilla Kindle firmware has it too. That would let you forget the part of connecting your device to a PC.
[0] KOReader also has an RSS reader, but I'm not sure how good is it and what it can support in terms of feeds "complicatedness"
I use KOReader's RSS reader, I haven't run into any issues yet. It supports RSS and Atom, supports pulling the full article text if it's in the feed, supports images, and puts each feed entry into its own epub.
[0] KOReader also has an RSS reader, but I'm not sure how good is it and what it can support in terms of feeds "complicatedness"