I also have tulips as well as other spring bulbs planted with strawberriesl. They seem to co-exist quite happily.
I never worry about picking off the flowers either. If letting them all flower and bear fruit in the same year as the spring they were planted was supposed to stunt their growth, they didn't get the memo.
And June bearers ARE stronger spreaders. Once their fruit production is done, they are pretty single-minded about taking over the world.
So, if you're concerned about strawberries spreading too much, then -- well, what I have experience with is -- Dayneutral Tristar is the way to go. I don't know if it's still true that Tristar is the better choice for mid-Atlantic/Northeast as compared to Tribute for the western states, but that was the assessment years ago. Other varieties of Dayneutrals have been developed since. You get MORE berries all at once with June bearers so it works out better to plant both, but in separate beds.