With services in particular you indeed don't need one.
No need to ensure the software is up-to-date, no worries about hardware outages, no need to monitor the system, etc. etc.
The person working with a rented service infrastructure can focus on what makes them money: the process.
With self-owned hardware you need contingency plans, setup, maintenance, redundant ISPs, UPS, extra electrical setup (many offices automatically turn off power to all outlets after office hours to safe electricity and reduce the risk of fires), and much more depending on the required tasks and their importance.
If you want to know what happens if you cheap out on that because "it just works" anyway once it's been set up and operated by an underqualified and overworked person, look no further than the recent ransomware incidents.
Some of them could've been easily avoided had the IT infrastructure been up to snuff. Way too many businesses and public entities still treat IT as a necessary evil that can be handled casually, not realising their dependence on it.
The specifics differ on a case-to-case basis, but in general it's a terrible idea for a small non-IT business to operate their own IT infrastructure. The solution doesn't have to be "cloud" or SaaS either. Hiring a competent 3rd party to deal with the specific implementation and operation for you is just as good even said party is "Bob's Business IT Services" and not AWS or Microsoft Azure.