aamoscodes had some good suggestions, and bunny.net would also work as well.
If you trust your VPS capabilities, you could also run a Peertube instance. It should have the embedding features you want. However, I'm not sure what sort of performance you are going to want on the VPS to make it close to something like Youtube. (How many TB of video do you have? How spread out is that 4 TB of stream data? If you do run it, I'd suggest running it through Cloudron to simplify security and backups (it's very good at that).
If you trust your VPS capabilities, you could also run a Peertube instance. It should have the embedding features you want. However, I'm not sure what sort of performance you are going to want on the VPS to make it close to something like Youtube. (How many TB of video do you have? How spread out is that 4 TB of stream data? If you do run it, I'd suggest running it through Cloudron to simplify security and backups (it's very good at that).