Background: I haven't been up the chute on Whitney since May 2000, but have been up Shasta multiple times in the last 3 years via Avy Gulch, West Face and Hotlum-Bolam Ridge (but I haven't done the Hotlum-Wintun Ridge).
IMHO: Avy Gulch is similar to the chute, maybe a bit easier if you stay climbers right toward The Thumb and cross the bergschrund above Red Banks. The chimneys through Red Banks are a bit steeper, especially the left one.
If you want a moderate climb without the crowds, hike over to Hidden Valley and climb the West Face (my favorite route on the mountain). It may involve a bit of route finding unless there is a boot pack from Horse Camp to Hidden Valley, but Google maps shows the summer trail if you zoom in and you can grab waypoints off it to help navigation. The camping area at 9200 has awesome views and usually a fraction of the people on Avy Gulch. The summit day is 1000 feet longer than Avy Gulch from Helen Lake but it's well worth it. West Face is less steep if you stay climbers right and then traverse back to the middle - the gully directly up the lower slopes of the West Face is 40+ degrees and can be quite icy for an alpine start. The route joins up with Avy Gulch at the bottom of Misery Hill (13,100 ish). If conditions are right, you can glissade 3000+ feet back down. My June 2016 climb is on youtube
I don't know much about Hotlum-Wintun other than it's rated D2 (same as West Face, while Avy Gulch is D1) and the Brewer Creek TH is often still snowed in long after the south side routes are open.
For any routes on Shasta, check the forecast, carry ice axe and crampons and know how to use them before going above 10K. And like the chute on Whitney, don't glissade unless you've tested the snow and know it's soft (which is usually not before 11am on Avy Gulch/West Face). [off soap box
Shasta Ranger station, The Fifth Season and Shasta Mountain Guides and Shasta Avalanche Center are all great resources.
If you want to be ultra conservative, do a 3 day West Face climb with Shasta Mountain Guides and you'll learn a lot from their awesome guides.
Good luck and have fun (if it's not obvious, I love climbing Shasta)