Friends, after ages of searching and searching, I have finally found it!
Here's a picture I posted to my site -- scroll to the bottom of the page.
Picture of Thunder Force 1
Even better, I found a page that offers an emulator, with the game, for download. Be warned, it only runs in DOS, but it should run fine.
Thunder Force 1 download
Now here's the story. Technosoft created Thunder Force 1 in 1984 for the Sharp X-1 computer. It was then ported to the NEC PC-88 computer, which is the version we have here; I understand the Sharp version looks better, but I still can't find proof. Anyway, it was also ported to the NEC PC-6001, the Sharp MZ-1500, and the Fujitsu FM-7.
Thunder Force 2 updated the overhead stages from the original, but added all the crazy special weapons, as well as the new side-scrolling stages. It first appeared in 1988 on the X68000, and then ported a year later to the Genesis, minus some levels for some reason (Grandia, Sol Feace, and Arcus Odyssey are also X6800 ports). The rest is history, as we all know.
I still haven't played TF1, because I don't have MS-DOS on my WinXP computer. I've also seen emulators for the Sharp X-1 and the later Sharp computers, and even got one to work, but damned if I could figure it out. Perhaps you may have better luck if you Google around.
Anyway, we at least know the story about Technosoft's first shooter. If anyone has something to add, by all means, please do so. I hope you enjoy.