My old speaker fell off my bicycle and got run over by a minivan so I got myself a new Bose SoundLink. It's got great sound and a strap to secure it to my bicycle. I used to be a headphones person a few years ago but now I listen to music instead of podcasts so I really enjoy the speaker.
The title asks what I look for in a Bluetooth Speaker, not what I think are the important features. Answering the title; what I look for in a Bluetooth (cycling) Speaker is the OFF button.
See that. Riding with my music brings me such joy that I completely dismissed that some people just like the sound of the street. Not me. I'm a music-rider-guy. I used to listen to podcasts w/ headphones, but I noticed that after my ride I had no memory of what the podcast was about. So if I'm not going to give it my full attention I'll choose music as my background/soundtrack to my ride. Maybe I'll try a few miles without anything playing.
From time to time, I've used my cell phone at top volume. I have a small bag that hangs from my handlebars that I can put my cell phone it and hear music. It doesn't blast others; it is ok to sing a long for most stretches; it totally does not work when underneath the el.
(I have found audiobooks from the library to be a great alternative to the radio/podcasts. I have listed to quite a few books over the years, and while I was initially a skeptic, I've grown to love them.)
Not sure if this would work for you but earbuds with transparency mode might be an option. You can play music but still hear sounds around you.
I have Jabra’s and use them at the gym but not for riding. I really like them.
Clear loudness. In the summers, it clears the way on the Lake Front Path when pedestrians are walking on the bike lanes.
I tried my SoundLink on the bike, but found it difficult to hear over the street noise. Plus I don't like pushing my music on the people around me. And I don't trust that strap.
I like my Jabra Elite 25e headphones. I know a lot of people don't like headphones in general while riding, but I've never had an issue hearing cars approaching behind me.
But if you want to be more aware of your surroundings, maybe check out bone conduction headphones (https://www.reviewgeek.com/5631/the-best-bone-conduction-headphones/). I tried some in a Fleet Street store and they sounded pretty good.
The op asks what we are looking for in a bluetooth speaker. I see posts at the extremes- wear earphones to don't listen to tunes. The former is dangerous on a bike and the latter is a lifestyle choice that has already been made by the op. So, let's have at it and advise about speakers. I occasionally, but not always, use one. Yes, a speaker does contribute to the sonic environment and yes, we need to find balance when we use them-- loud enough to hear and soft enough to not be that guy/gal. I have a Boombot from Boombotix. https://boombotix.com/. I did not pay a lot and it works decently. The clip is great if you are using a shoulder strap for a bag. However, it doesn't hang great simply on a shirt or jersey and I have rigged it up to the brake cables facing me on my handlebar. I used to use it at home but was recently gifted with a JBL Charge 4 for the house. There are plenty of other portable speakers out there. What are they?