I'm looking for an alarm clock ringtone that isn't loud/intense. I rather gradually wake up instead of jumping when it goes off.
I've used some of the ringtones that were provided by Apple but didn't care for them. That was new and classic tones. I don't see the stock bedtime tone.
"Bedtime" isn't a tone, it's a submenu within the Clock app. At the bottom, instead of choosing Alarm, choose Bedtime. You can set a wake time, and a Wake Up Sound, and the sound starts out really softly and increases in volume if it's not Snoozed or Canceled. I really like the "First Light" tone...it's a very gentle wake up.
I see you have a Apple Watch. If you are looking for a tone that isn't intense, the Apple Watch fits the bill. Why aren't you using it?