Some of Giant’s bikes, such as this Anthem X are specced with Shimano press-fit bottom brackets and XT or SLX chainsets. They don’t give model numbers for these bits but CRC and other online retailers only seem to sell HTII XT chainsets, there’s no press-fit-compatible Shimano chainset on display. So does this mean that press-fit bottom brackets are compatible with normal HTII chainsets? And by extension then these frames and bottom brackets should also be compatible with Race Face X-Type chainsets.

Is that correct or am I horribly mistaken?

You are correct.3 points.