I think there are two main reasons why felt picks aren't more popular than they are. First, they tend to wear out quickly and leave behind dust. Second, and probably more important, the people who like them usually do so because they are trying to approximate the sound of fingerstyle without actually playing fingerstyle (for whatever reason), but most people who favor picks do so because they want a tone that is different from fingerstyle. The ideal, of course, is to be equally proficient at both styles and switch back and forth depending on the song (or other factors, like fatigue).Yes I've seen quite a few pick threads since I started on here, and some mention the felt sort. I get the logic of that, I'd be interested to see if they do make that much difference, because if they did they surely everyone would be using them? Assuming, of course, that everyone has the same problem with the tinny sound!
Two other thoughts: First, try turning down the highs on your bass (or amp or EQ pedal) when using a pick to cut down on the "tinniness." Second, note that the pick tone will sound much better when playing with a full band than when playing alone. It's just one more example of how things that sound good in the bedroom often don't sound good on stage, and vice-versa.