From a purely financial point of view it is never beneficial to outsource your infrastructure, and I'll explain below. There can be secondary benefits that make it better, but it's never the actual infrastructure itself.
Lets say you buy $100K worth of infrastructure, you can then take that as CapEx and budget it over it's ~5 years worth of life expectancy at 20K per year while also taking depreciation as a tax deduction. If you want to migrate your infrastructure to "the cloud" which is just marketing slang for "another mans datacenter", then you will be paying yearly OpEx. Other mans datacenters are just leasing you time on the exact same 100K worth of infrastructure, except they are going to oversubscribe at 2:1 at the best price tier, 5:1 or worse on the lowest tier. They are doing the exact same CapEX + depreciation that you would of done, only now they are charging you $50k/yr to you and two to four other people to use that $100K worth of infrastructure. You will always spend at least double if not triple the amount of money over that 3~5 year lifecycle period, possibly even more since some places will keep low priority infrastructure around for 10+ years. And this is before the tax benefits of using the CapEx + Depreciation model over the OpEx one.
Where "another mans datacenter" can help is in variable loads or multi-region access. If you have an application that runs low utilization for 20 hours in a day but then does insanely high utilization for 4 hours, then you can go cheap for the 20 hours and then pay more for those 4 hours. Also running datacenters, even CoLo's in multiple-regions gets expensive and kind of silly, abstracting it buy renting infrastructure capacity in "another mans datacenter" can make that cost more manageable. The final benefit is simply not having to manage physical infrastructure, which is invaluable for software startups. If a company is just getting going or is super small, they need more flexibility then the 3~5 year infrastructure lifecyle offers, and renting it can come out better.
The economics never work out unless you are seriously overbuying on infrastructure, we're talking 2~3x more then what you need. This is why everyone is moving back to a hybrid model of on-prem or leased rack space with some functions in "another mans datacenter". Equinix has this really cool model where they'll run fiber from a port outside your rack into AWS, Google, Microsoft or any of a couple hundred peoples infrastructure. You then do BGP peering and viola you can directly communicate between your infrastructure and the other service provider infrastructure without having to egress onto the public internet. You can run heavy processing and data on a system you own while having it accessed by a "cloud" service in the single digit millisecond range. Was really cool to see a bunch of EC2 instances running on private storage.