The credits following a recorded video performance are both customary in the industry, and are also required by union the union contract that applies when the production company is a union shop. These customs have not been static, however, and the history of this particular practice (without its underlying legal context) is discussed in a relevant Wikipedia article.
Almost all professionally made entertainment video programming in the United States on television and in the movies, other than pornography, is covered by union contracts, because actors, screenwriters, producers, directors, electricians, costumers, etc. are almost all part of unions or guilds or trade associations organized on an industry basis (as are almost all regional, broadway and off-broadway live theater performances, and as is most live mascot work). See, for example, this list of associations.
The Screen Actor's Guild is representative of how this industry-wide union system came into being and works today. In a nutshell, Guild members agree not to work for any production that does not have a union contract, on pain of being kicked out of the Guild, and any production with a union contract is forbidden from hiring anyone outside the union to a screen acting job within the Guild's jurisdiction.
Guild members have a strong incentive to honor this requirement because (1) an actor who is kicked out of the Guild is not eligible to work in union productions which have a 97%+ market share (all of the top 79 grossing movies in the U.S. in 2015 were unionized and at least 97 of the top 100 grossing movies in the U.S. in 2015 were unionized), and (2) most of the employee benefits that more conventional employees get through their employer, like health insurance and retirement plans, are usually secured by Guild members through the Guild. Many screen actors also rely on the Guild for referrals for new audition opportunities.
This system was originally organized in the 1910s and 1920s and has remained in force pretty much ever since then, and is organized similarly for each of the kinds of employees that a production company hires.
Generally, the video productions of a non-profit youth ministry would not be within the jurisdiction of any of the industry-wide union contracts that mandate that credits be given for performers and back stage people, particularly if they are volunteer productions. If you don't need to hire union actors, videographers, etc., you are not legally bound to comply with the terms of a union contract.
It also isn't unheard of for a union performer to obtain a special dispensation from their union to work on a volunteer basis, or a below market rate basis, for a particular isolated engagement with an organization such as a non-profit youth ministry. Even then, however, this dispensation would often still require that the non-union organization using the services of a union member at least honor union regulations regarding crediting the people involved in the performance (about which the various unions within the industry have reached a cross-union consensus).
Usually, however, any license to perform a particular work would require, at a minimum, an oral or written statement along the lines of "By arrangement with Bill Holab music" (that particular one appears on the program of the opera performance I saw yesterday) that credits the copyright owner, in connection with any performance of the licensed work, even if there is no royalty due or the royalty is only a one time nominal $1 or $10 payment.
Also, many non-U.S. countries incorporate a right of attribution into their copyright laws (these are called moral rights), so if you are producing a foreign copyrighted work, whether or not it is under a license, to be technically squeaky clean, it would be necessary to credit the author of the copyrighted work (e.g. the screenwriter and composer), even if the author no longer owns the copyright. The odds of ever facing legal action for a violation like that, however, are remote, and performers and stage hands generally do not have rights of attribution.
Lots of pornography is non-union, but it is subject to a requirement to verify that its actors are adults and include a certification to that effect in lieu of actor credits that would be required in a union shop. Unless your youth ministry produces pornography (in which case, you have a variety of much more weighty legal issues to worry about), you don't have to worry about this requirement either.