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"
[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"