My church has not outsourced it's IT Department to a Managed Service Provider, but my previous employer (family owned retail stores 120 employees, three locations, 50+ computers) did move to an MSP with some level of success. The amount of manpower, skillset, geographical deployment and time coverage that they could marshall for specific situations compared to onsite staff was no comparison.
The biggest struggle that we had was who set the vision and direction for the relationship. The business owner specifically asked me to be the onsite 1st responder, with the MSP to be brought in at my discretion. I was also asked to set the goals and manage the majority of the business's interactions with the MSP when it came to projects. The MSP had a completely different perspective and believed that their primary responsibility was only to the business owner.
Assuming the vision and clarity of who does what in the relationship is clear, I believe that MSP's are a viable option. Do keep in mind that having someone geographically local (less than 15-20 minute drive) is immensely helpful. If your network or phones are down, having someone who can only provide remote support means nothing if they can't reach your equipment.
Finally it all depends on the skillset available in your church, if you have no one who has a clue about IT - then an MSP is wise idea at least 1-2 years to modernize your infrastructure and give you a chance to hire someone on staff and get them trained.