From a design perspective there are two main reasons why we would recommend or require "landing at nearest suitable airport" or similar wording.
One is that the aircraft has so degraded performance in one or more systems that we cannot assure continued safe flight and landing for other than a limited time. And example of this would be something like multiple electrical generator failures resulting in having to rely on battery power.
The other is that the failures themselves do not cause any immediate or significant reduction in systems performance, but DO result in degraded system redundancy. An example of this would be loss of hydraulic power in one system; most aircraft have sufficient capability to cope with a single hyd failure (if designed with multiple systems) but loss of a second system would be far more serious. So we want to get you on the ground ASAP to minimise the time of exposure to a second, far more serious, failure.
Unless you have been very unlucky, most failures are of the 'reducing redundancy' type, so most diversions mandated by the AFM/procedures would be precautionary. If nothing else went wrong you probably could have continued on to your planned destination; however, you're already having a bad day; why press your luck?