Maybe he doesn't want to? Not every musician is bursting at the seems with ideas they want to write. Sometimes they just love the act of performing. There's been some instances of him being told he can't write lyrics to a song here or there in DT, but I'm pretty sure in his solo band, the dude can do what he wants. So if he's not writing lyrics, he's probably not fighting terribly hard for it.
I always had the impression that LaBrie kind of wasn't allowed to write a lot of lyrics because he's never (or very rarely?) involved in the song writing process and the lyrics and vocal melodies are already written when he comes into the studio to start recording vocals. However, I have no real evidence of this. I didn't mean to imply that he should write lyrics. If LaBrie chooses not to then that's cool.
No, it's not really a question of "allowed to" or not. James is perfectly welcome to write lyrics. I honestly think that if he came into the writing sessions with a bunch of ideas and said, "Hey, I have a lot of lyrical ideas that I think would be great for 8 of the 10 new songs on our new album," JP and the others would likely react along the lines of, "Great! Have at it!"
I doubt it, personally. I think JP is much more of a control freak than one might think because he is always relatively quiet and humble sounding. I mean, when you have written the majority of the lyrics for every release, have also been the main songwriter and producer, I doubt youd be willing to give up that much creativity.
Oh, he definitely is a "control freak" of sorts. When you talk to him, he makes no bones about it. But that said, he is also VERY open to others' ideas and input, and I have gotten that not only from several conversations with him, but with others in the band as well. It kind of depends on the issue. For example, on the issue of having Jordan take up guitar on a song here or there, JP is pretty inflexible and doesn't want that in the band. If Jordan ever
really wanted to and pushed hard for it, might he reconsider? Perhaps. But as it stands now, he has been pretty clear that he is dead set against it. But then take the recording of the vocals for ADTOE. He wanted everyone together in studio. But James wanted to record the vocals alone in Canada. JP gave him the complete green light to do that. The one exception was the vocals for BTS, because he felt very strongly about how the delivery should go, so he took a trip up there to give input on that one. But otherwise, he deferred to James.
How would that translate to lyric writing? Obviously, none of us know
for sure. But given what I know of how the guys work and what they have said to me on other topics, I still strongly believe that James (or anyone else in the band) could take a much larger role if they wanted to. I mean, I doubt any one person would write the vast majority of lyrics on an album because JP likes doing it and wouldn't completely step back (I was exaggerating when I said 8-10 songs). But I really don't think he would have any problem with James taking an expanded role if James strongly felt he had more lyrics he wanted to contribute.