This is what I did. When the Google News redesign happened, it made Google News substantially less useful to me. Enough so that came up with a replacement.
It's not for everyone, as it requires running your own webserver, but I use Tiny Tiny RSS to aggregate the feeds of the various sources I'm interested in, then can read the aggregated feeds (I have multiple, a different feed for each general subject) through the web interface and/or by using an RSS reader. I use an RSS reader (gReader) on my mobile devices to do this.
It's not for everyone, as it requires running your own webserver, but I use Tiny Tiny RSS to aggregate the feeds of the various sources I'm interested in, then can read the aggregated feeds (I have multiple, a different feed for each general subject) through the web interface and/or by using an RSS reader. I use an RSS reader (gReader) on my mobile devices to do this.