The routine I mentioned above is my standard routine for most of the year. What I have learned over the years is that rucking and time on feet helps a lot for the Grand Canyon. I'll wear my backpack to walk my dog which gives me an extra 10-12 miles a week of low impact work. Read an article from a SAR volunteer for Mt Whitney an they said where most folks get in trouble is that they train with a 5-6 pound pack but on game day they are now carrying triple that weight. The extra weight causes issues with fatigue, which leads to other issues.
One other thing, one of the guys that hiked my R3 with me last year is a food scientist at one of the name brand energy bar companies. He's helped concoct some of the bars and gels that you probably eat when training. Dialing in your nutrition is probably the most important aspect of a successful Grand Canyon trip. His take is that you need 250-300 calories per hour, every hour starting from hour one. Then divide those calories between food you drink and food you eat. For many people, if you can hike/run 4-5 hours you can hike/run 20 hours in the Grand Canyon with proper nutrition and dialing back your intensity a tad bit.