Pros & Cons - I guess it would be interesting to discuss..
Snapshots are 1.5-2$ per 30GB (iirc)
Run databases in Managed RDS (huge ++ and if we can add MySQL profile using Aurora Serverless, it would be double awesome. Haven't tried it)
On the fly resource upgrade
Easy maintainability if you have the know-how (even if you forgot the root password ?)
Hardware change is done by a reboot (transparent to you during any maintenance announced by AWS)
Add multiple levels of security and lots of compliance associated with AWS is yours too.
I have seen two relatively big web hosts, move their infra to hybrid, both Private Datacenter + AWS infra. It can turn out to be expensive, so be cautious. You pay extra for each GB of storage and each GB of traffic.
Cost is cheaper and fixed too (is a bundled package as well)
Check for DDoS protection and not everyone has it.
Dependent on datacenter technician to replace the hardware.
Need 24/7 monitoring for your hardware, even RAID failures.
You need IPMI for console access.
Limited resource expansion
Offline Backup has to be managed
I also noticed that Jelastic (another provider of cloud infraestructure software) has created CPanel templates that have the advantage that their CPU allotment is scalable (up and down automatically) and you only pay for the used disk space storage however I have some doubts about its performance as it run as a Docker template.
As @SS-Maddy said, I think cost is going to be the biggest factor. If you can run on Lightsail, then your costs are pretty much fixed. If you run on regular AWS there can be a lot of other fees that are hard to estimate (until you have been running for a while) and can really drive your costs up. Most dedicated servers are a fixed monthly cost.