Wow, $1 for 100 emails is very very very very expensive. So expensive that I'd opt to build this in house rather and take on the costs associated with it and still come out saving money. It's a pretty solid idea otherwise.
Approvals here cost ~1¢. Picking a figure of $100k as a developer salary (as it was the first result in a quick search) that puts a lowish bound as about $400/day. So you could take your estimate of build & maintenance times per year, multiply by about 40-50,000 and that's how many approvals you're looking at.
So if you're google doing this every login, sure, that would be prohibitively expensive. If it's my accountants site doing it before they charge me a different amount for filing (there's one of these every year) or being the approve/reject part of an HR holiday system, or anything less common you'd have to be pretty big before it'd be worth spending a day or so building it yourself.
Personally I could see myself using it in some automated workflows I'm building now. Script runs, but before it does something irreversible (e.g. ordering an item) it checks with me that things look OK. It'll cost me a tiny fraction of the total spending, and it's ready right now (and works!).