California does have the tallests mountain in the contiguous US. It sure as fuck isn't Mount Shasta.
Mount Whitney stands 14.5k feet tall, and it's way more fucking badass in Appearance.
Many of Colorado's mountain summits also stand taller than Shasta, and are multitudes more majestic in appearance.
As for iconic, the Appalachian mountains may not be known by individual name, they are the boy band of mountains in terms of fame in the US. Mount Washington is also extremely iconic. I hadn't even heard of Shasta till this post. St. Helens is also extremely well-known.
For those wondering, Denali is the tallest mountain in the US, as well as the tallest mountain on land in the world.
EDIT: so for clarification - Mount Everest is the highest mountain in the world, at close to 9k meters. When you measure it base-to-peak, however, Denali measures out at 5,486 meters, while Everest is only 5,200 meters.
Something notable is that though Denali is stated as the tallest mountain on land, it seems Rakaposhi in Pakistan has a base-to-peak measure of 6,000 meters. It also is the only mountain whose peak descends to base without interruption apparently.
EDIT 2:got my numbers mixed up, fixed some info.