The problem with buying engines as part of the airframeis the ancillary cost of maintenance (scheduled, unscheduled, catastrophic events, downtime, spare parts storage etc). You carry the cost & risk.
Instead you buy or lease the airframe & contract with the powerplant supplier to provide a guaranteed source of power for your airframe. They carry the risk for unexpected maintenance etc. and will pay the cost for the maintenance required to ensure a working powerplant(s) is always available on the aircraft.
When you operate a small number of engines the risk of unexpected maintenance requirements (particularly catastrophic ones that require, say, an engine rebuild) is relatively low. The problem is that IF it happens the cost might be unaffordable.
When large numbers of engines is operated then the risk is relatively higher, however the cost of a single event is spread across the relatively larger maintenance organisation. Proportionaly it is less 'costly' eg carrying one spare engine is a small(ish) cost when you operate 500 engines, not so if you only operate 2.
The power-by-the-hour organisations enable you to have similar scaling effects of a large operator.
Of course you are also paying for the profit margin of the power provider so there are some extra expenses in the transaction but depending on circumstance it still be cheaper overall.