Did Mt of Holy Cross 2 years ago. Nice hike, great views, lots of vertical, but even with 2 other hikers, it was VERY creepy. Many stories about hikers disappearing on trail never to be seen again. Even local bars and eq shops willing to expand on that, etc. Googled, and is true. How adventurous are you?
I have spent a lot of time hiking/running in Colorado. Have climbed Holy Cross twice and never heard of these stories.
Nor did I ever feel creeped out. On the summit ridge my partner and I experienced a PR for electricity, but that could have happened on any peak/ridge that time of year. Anyway, it's close to neither RMNP nor Aspen so is a rather moot point WRT the original poster's request. (Otherwise I'd be another singing the praises of the San Juans...)
My favorite hike near Aspen is the "Four Passes Loop" which circumnavigates the Maroon Bells. The four passes are West Maroon, Frigid Air, Trail Rider, and Buckskin. It's about 28 miles and can be done in a day by a strong party, comparable to the Pemi Loop but a lot smoother trail. Check snow conditions if early-mid summer (an ice axe may be useful in spots). Many people backpack it over 3-4 days.
If you're looking to do 14ers, the Bells, Pyramid, Snowmass, and Capitol Peak are pretty gnarly, Class 3 & 4 hikes which your non-technical friends probably would not enjoy. Castle-Conundrum are the mellowest 14ers in the Aspen area. Fortunately, lower elevation options abound.
If you're spending much time hanging out in Estes Park and Aspen, plan on spending some $$$. Personally Leadville, about halfway between the two, is more my style.
Have fun!