He's sort of right... not about Half-Life, but its expansion.
It's true that Half-Life was developed for PC, and was actually released a year before the Dreamcast's US launch. When Gearbox ported Half-Life to the Dreamcast, they developed HL: Blue Shift as a DC-exclusive campaign. Of course, the Dreamcast version went unreleased, so Blue Shift was later released for the PC.
Gearbox also created higher quality versions of many of the game's character and weapon models, which were advertised as another Dreamcast exclusive. These models were slightly enhanced and released for the PC as the "Half-Life High Definition Pack."
It is known that the PC version of Blue Shift is a port of the DC version. For their port, Gearbox used the same maps, which aren't 100% compatible with the PC version of Half-Life. Not a problem initially, as Blue Shift was packed with a modified version of the HL engine, which meant all Blue Shift's special features were left intact. When Valve released Blue Shift on Steam, they packaged it with the vanilla Half-Life engine, leading to missing effects (Blue Shift's special HUD) and numerous glitches (one notable error made progression impossible for many players.)
Gearbox was behind both the DC and PS2 ports. You're correct that it isn't known whether the PS2 version uses any code from the DC version. However, some of the graphics in the PS2 version began their development on the DC. Essentially, the PS2 version uses enhanced versions of the PC's Half-Life High Definition Pack models, which began development on the DC. This is most noticeable with the weapons. For the DC version, Gearbox replaced the Glock and MP5 with the Beretta and M4 Carbine, respectively. This change was maintained for the PC's Half-Life High Definition Pack and the PS2 version.
In short, it's fair to say that Half-Life's high definition graphics began their development on DC (where they went unreleased), but made it to the PC and PS2. The same can be said of Blue Shift, which went unreleased on DC, but made it to the PC.