I've been looking around for a while, but I've never been able to find anything good that isn't a parody song. I'm hoping to find something on the same level as these: Surely a fandom as large as Star Trek's has produced something of equal quality?
There's a group called T'Pau named after the Vulcan High Priestess from "Amok Time". Mostly their music doesn't have anything to do with Trek directly, although there are a few tracks that feel Trekkie if you squint your ears. Especially this one, which always puts me in mind of the asteroid mining operation in "Spock's World"... Brent Spiner has an album called "Ol' Yellow Eyes Is Back" that is him doing a good job covering Sinatra-era tunes, and some of the other TNG actors back him up. And I don't think of this as a parody, so much as maybe filk?:
Well, Richard Whiting's song "Beyond the Blue Horizon" was always credited by Alexander Courage as the main inspiration for the Original Series theme tune, can't find a sung version, but here's the music- The theme tune does actually have lyrics written by Roddenberry, covered here by Tenacious D- Aaaand of course Shatner sings "Rocket Man"- You're lucky I didn't link to "The Ballard of Bilbo Baggins"...
I've got a different version of this that came on a CD with disco versions of different sci-fi themes. I think I prefer this version. The other version has the lyric, "We'll star trek the unknown." Which I always thought was a little cheesy. Although it's the first time I know of that Star Trek was used as a verb, long before Star Trek Into Darkness.
For me, personally, I suspect that the kinds of songs I like don't exist for Star Trek. It makes me wish I had musical skills so I could write one.
You are, I fear, most certainly correct. Star Trek lends itself to parody and comedy songs much more readily than meaningful ballards, alas. I think your best shot is to look for existing songs which our esteemed contributors here feel most captures the spirit of star trek.