The cloud is only cost-effective if your workload fits into their loss leaders where they're actually giving you reasonably-priced services. Anything else and you're still getting scammed. It may be that the amount is too low to matter and the pragmatic decision is to keep going for the time being, but you're still getting scammed.
The fast scaling/up down is a good argument but you need to proceed with caution:
1) is your application actually structured to be able to do this? Would the overheads of developing your application in this way negate the savings (a fully distributed service with queues/etc has more development/management overhead than your basic monolithic MVC app, which may eat up all your savings)?
You need to do your own numbers here and be pragmatic. You'll have plenty of resume-driven people who see the potential of an engineering playground and will immediately tell you whatever it takes to make this happen, regardless of whether it will benefit the business.
2) how much does it actually cost to run your application at full scale outside of the cloud? I've seen places that do the whole autoscaling thing (and have lots more moving parts as a result) and yet a fully-scaled-up equivalent on bare-metal would cost less than the baseline scale on the cloud, thus removing the business case for it.
Keep in mind that the cloud effectively ate all the hardware improvements we've had in the last decade so switching to bare-metal hosting means you'll need a lot less hardware. A cloud "vCPU" is more like a pocket calculator when compared to a real CPU, and same with storage.