Having worked at one of these for ~6 years ... no they don't (well, it is true that you can get extra services of course, and I bet there are some that do charge, but in general, they don't).
For having a hosted + backed up database that's redundant (I hear master-slave with auto-promotion is quite common, mysql and postgresql are available) and that THEY will fix when it becomes unreachable, or cable cut or ... that's a standard product, and there's no fees for fixing it when it goes down.
All I know is that the most expensive dedicated customer we had (which we did a database for) paid less than $20k, and he had about 800000 DAUs (and is a well known site).
Also, in all the time I've been in the industry, I've known a single instance of hardware failure leading to data loss. I've heard of dozens of times of perfectly authorized accidentally erasing/corrupting the database (mostly because they wanted me to figure out how to repair it). This is the real threat, and will require consultancy to fix, given the average level of technical competency of cloud customers, both on cloud and dedicated.
Another one was a dating site that had less than half that in charges (no idea about DAUs), moved to amazon, balked at a bill that (after serious optimization and large customer discounts) exceeded $100k mostly moved back to us (only using amazon as backup location).
Cloud has a number of advantages, but price is definitely not one of them. And there's serious "fine print", like that most of the cloud advantages (like "not losing data when a physical machine dies") don't apply to small customers.
The big secret is that scaling customer apps generally runs into issues because the customer's programmers are not considering efficiency at all. Letting me loose on their code results in 20-30 TIMES improvement within hours, not because I'm that good, but because usually the first factor 10 is simply not getting the same data 10 times from the DB. No real difference between languages in the programmer's skill level (with exceptions for rare languages : Haskell programmers, for instance, are definitely better, but that will stop if it ever becomes popular). There are exceptions, of course, but this is the common situation. Having done 2 cloud migrations in the last year I worked at one of these managed hosting shops, scaling doesn't work any better on cloud, in fact it often works far worse because PHP's performance far exceeds what you get with ruby or python/django. When you exceed shared serving you can move up to managed/dedicated for a big performance boost (in fact the hoster will probably do this for free when you upgrade your plan), which is likely to last you for more growth. The issue is that PHP shared hosting in my experience often serves more customers than 4-5 dedicated django or RoR servers can (mostly because like all ORM, django's ORM means 40-50 database calls per page showing, RoR is no different). And unlike cloud, serving pictures or even video will result in your site slowing down, not in a $10k bill.
That said, there are things the cloud is good for. We fail badly for customers that require having truly large amounts of diskspace simply accessible (we're talking 10+ Tb or so, that's constantly accessed, before this becomes an issue), or if they need massive amounts of compute that needs to scale with very little warning. Even in those cases, the times I've seen they quickly decided that storing some files in the cloud through remote filesystems is far preferable to actually running the site on the cloud, due to cost. AWS lambda is pretty useful for this: quickly run some code on input, store result on S3, don't keep the remote task running. Uploading java code is easy, so if the task is processing/tagging/creating reports/pdfs/... it's easy to code up, test locally, and packaged as a jar it will work.