Thanks for asking Jasmine.
Interesting question and I must admit I’m new into understanding how a church establishment works, but from your detailed explanation above, I believe I get the idea.
Instead of creating a new separate website just for your media ministry, I’d recommend “parking” it under your church’s main website. This way, you are combining your SEO effort into just one website, without having to work on the SEO optimization for two sites.
However, if you plan to establish the ministry as an independent one without having a close affiliation with your church, then yes, perhaps you should create an entirely new site. You could use an open-source software that is similar to YouTube or purchase a YouTube clone script. But remember that if you plan to build a site, you need a place to host your videos.
Or, you might want to consider building a YouTube channel instead of the put in the work on a new site. This way, you get more visibility, since YouTube could drive viewers to your videos. Furthermore, YouTube provides subtitle feature which makes it very easy to create subtitles. The easiest way is to use the auto-generated subtitles by YouTube, then export the transcript and perform human-translation work to correct the translation mistakes. There are human-translation websites like Amara or Gengo (disclaimer: I never used any of them) or even Fiverr (check the quality of your translator on Fiverr. I hired a Vietnamese pretending to know Korean language before) or you could find volunteers among your church members.
For curating content from other churches, you may engage someone to be the content curator and they will be in charge of reviewing and uploading the content on your channel.
Bottom line is, since it’s for a non-profit project, I’d recommend moving towards building a YouTube channel rather than build an entirely new site, unless you have compelling reasons to do so. You could always have both – YT channel and website with embedded YouTube videos.