It's not CalDAV, but Joplin supports syncing with WebDAV and works great with Nextcloud (or any WebDAV server, really).

Nextcloud is a love it / hate it kind of thing. Since you're not already running it, I assume you don't want to (fwiw, it also does Cal- and CardDAV). Filerun used to be the go-to alternative, but it sold out and is all paid now. I'm not sure what took its place, but at the very least you should be able to setup a basic WebDAV server using just Apache.

The only thing I don't like about Joplin is it doesn't have a web app (at least, not as of the last time I checked).

There was another note-taking app I was thinking of and about to suggest, but it syncs over IMAP.

I hate the fact that none of the big names support CalDAV natively. DAVx5 is cool and all, but app developers really need to step up their shit and support CalDAV already. Not just Microsoft Exchange and Google Calendar but CalDAV as well. It's not like they need to rebuild their apps from scratch.

At this point you might just be better served using a web app instead of a native mobile app. Maybe K-9 Mail transformation into Thunderbird Mobile might bring some good news, but I'm not holding high hopes.

Maybe we should, under the EU's DMA, force anyone that bundles a calendar/note app with their phone OS to support CalDAV as well as any proprietary protocol of their choice.

Maybe it's not what you want to hear, but I was looking these types of apps, too, some time ago.

Then I noticed that most of the apps are useless for me and found an interesting type of apps that implement a "second brain". There are a few of them, but I chose Logseq.

There seems to be a request to add this to the new Thunderbird CaldDAV features: https://connect.mozilla.org/t5/ideas/support-vjournal-in-thunderbird/idi-p/46295

KDE's KOrganizer supports journal entries

Which OS?

I've been searching for this for a long time and I can assure you there isn't any. jtx Board is the only one that works and is actively maintained (and unfortunately I hate its UI).

It's a pity because CalDAV notes would be perfect but unfortunately it's a pipe dream at this point.

Look for something that syncs with WebDAV instead, or has any other alternative method of syncing. For example there are apps that simply edit regular .txt or Markdown (.md) files locally in a folder, and you can sync them separately using FolderSync, Syncthing, Dropbox or any other method you want.

https://noteapps.info