If you're using Samsung's My Files, Long Press on the file & at the bottom you'll see the "share" icon, click on it & then you'll see the share options pop up, scroll down until you see the Bluetooth Icon & click on it.
Galaxy S20 Ultra on One UI V3.1
Your Note 10 on One UI V3.1 should have the same "targets" to share files etc. I'm pretty sure that my S10+ had the same "targets" & it wasn't running the same Android version as I have now.
Have you disabled any apps in Settings > Apps? It's possible that could have removed some of the options to share?
It might seem a little drastic, but you could try a Factory Reset & see if you can get back the same options for sharing as before.
While maybe not as convenient, as previously posted, you could always send the file you want to share to your Cloud account & email the link to the person.
I'm sure that you know that Bluetooth requires the other device to be within Bluetooth range as well.
Try using this app, "Feem" to transfer videos, files etc between your Note & the iPhone. You can transfer using WiFi.