Traffic that spiky is extremely unusual. But you don't need to maintain all that hardware capacity for 10% of the day.
There are a number of colo providers that also provide managed hosting and cloud services (even if you for some reason couldn't deal with the latency of simply tieing EC2 instances into a colocated or managed hosting stack via a tunnel).
In fact, combine, and you can cut the hosting costs even more than with colo services alone, since you don't need to plan for a low duty cycle for the hardware - you can run things on 90%+ load (or wherever your sweet spot turns out to be) under normal circumstances and fire up cloud instances for the spikes.
That method handles the loss of servers etc. as well just fine, again further cutting the cost of a colo/managed hosting setup because you can plan for less redundancy.
Personally I've yet to work on any setup where the admin costs of an AWS setup has been nearly enough lower than colo or managed hosting to eat much into the premium you're paying. You have 90% of the sys-admin hassles anyway, and you're left with far less possibility of tuning your setup to what works best with your setup.
Most of the setups I've worked on come out somewhere between 1/3 to 1/2 of the cost of running the same setup purely on AWS. Sometimes the gap is smaller if you have really spiky traffic or have lots of background processing or one off jobs where you can e.g. use spot instances extensively.
I do understand people wanting to pay to not have to think about these things, though. But you're almost certainly paying a steep premium for it, even with your traffic spikes.