Again this is not about whether BRD/MCH is currently viable or not. It is simply a discussion of how they are categorized and who they compete against for a spot.
Support exists as a sub-role from dps specs in SB. Bard and Machinist, together with them netting less dps than the rest...
The problem:
They achieved this with one (BRD), but not the other (MCH), simply because the latter utility is almost negligible.
The solution:
1) Remove MCH from that quota and bump their dps to the rest's level (most boring option).
or
2) Add more utility to MCH, perhaps one that you can keep up more often, like Bard's. (best option, some people genuinely love to play dps classes that support the overall group).
Technically, BRD is the support out of the two and MCH is more the DPS. Right now though, SE kinda goofed on balancing as, somewhat like DRG, the idea in their head and the reality in-game are very different.
Can MCH support? Yes, like DRG there are some tools to support the group. But BRD has many more as that's been their entire "thing" since 2.0 (old school Foes comes to mind, same with Mages when Healers needed MP).
They have always been a DPS job, thus they are in the category, DPS. They have raid utility yes in restoring resources, but they are still a DPS.
Other DPS have raid utility to boost damage in some form or fashion, but that doesn't classify them as support. The devs have said that there wont be a full/true support job as people expect, as that would be an imbalance to team composition, raid damage, and how the game works in general.
The closest you get, IMHO, would be Astro
MCH is not much of a support if you compare to BRD because of way too many skills.
BRD even deals more DPS with much less, but you CAN say BRD is a support dps because hell, they can even esuna or increase heals received by 20% along with many other res and crit buff etc.