The entire Copper Triangle is totally rideable w/o the event, the roads are safe. I think I've done it 4 times, doing it again in a couple of weeks. The ridiculously steep section is very very short, and can be walked if needed, I bet you won't be the only one. It is on the bike path between Vail and Copper.
That route is absolutely gorgeous, by the way. Don't psyche yourself out about it, tons of people ride it, and there will be support. Train for climbing, the distance, and altitude if possible, pace yourself when you do it, and have fun!
Mt Evans doesn't have anything steep, the crux is there's not enough oxygen up there.