I had this problem. My iPhone 13 connected to the Bluetooth device (ie: to speaker) but I couldn’t find the speaker option when playing Spotify, YouTube, Music, etc. It works now.
Here’s what I did: I connected to the Bluetooth device, in the device list I tapped the ‘i’ incon next to the device and selected ‘speaker’ (options were Car Stereo, Headphones, Hearing Aid, Speaker, Other). Then I unplugged the Bluetooth and restarted my phone, reconnected to the device, and now the speaker option is found in Spotify, etc.
Maybe it’ll work for you?