As stated by Le Monde[1] Kohlchan was hosted on OvO Systems, which was owned by Vincent Canfield. If you can't read the end of the article due to the paywall, you can use the reader mode of Firefox.
You can also see that cock.li is hosted on OvO Systems on the footer.
From what I understand OvO Systems runs a VPS host (cockbox.org) and after Krautchan shut down, Bernd rented a VPS on cockbox (since they allow anonymous payments) to host a Krautchan successor named Kolchan. So yes, OvO Systems hosts it, but the admin is not actively involved, similar to how the Hetzner admins aren’t involved in Plex movie warez services being hosted on Hetzner dedis.
Cock.li on the other hand is run by the OvO systems owner himself.
[1] https://www.lemonde.fr/pixels/article/2019/10/11/derriere-l-...