Drinking/administering a potion takes an action, as you cited from DMG at p.139. That's the general rule in effect, unless specifically contradicted.
You note that the DMG's description of the Potion of Healing doesn't mention that it requires an action to consume, and seem to be wondering if that omission is meant to signal something. But every other potion described in the DMG also omits any description of the mechanics of drinking potions: they just start with phrases like "when you drink..." or "for one hour after drinking...." So we can't read anything into the omission, or the general rule would apply in no cases. (DMG5e pp.187-8)
So why does the PHB call out the consume action in its Potion of Healing description?
The Potion of Healing is the only potion described in the PHB (p. 153); the PHB doesn't have a general "potions" section akin to the DMG's on p.139. So this is the natural place in the PHB to mention that interaction with the action economy1. Would it have been clearer to add to the sidebar on p.190? Maybe, but that's a sidebar full of free interactions, not examples of things that aren't-free-interactions-though-you-might-think-they're-just-the-same.
Sidebar: But I can quaff an entire flagon...
Yeah, this bugged my table too. We made in-world sense of it by saying that a magical potion is an... unusual enough mouth-experience to take a moment2 to consume. A flagon of ale... apparently the game designers assume our characters are more-inured to that experience than to the experience of drinking spells.
1 - it's also mentioned in the Herbalism Kit, but that's the least-natural place to talk about the potion's consumption.
2 - I think about trying to get my kids to take emoxicillin and Wish it only took six seconds =P
Cooperative Crafting increases the value of what you are crafting per day, not the amount of things being crafted in any given time, nor the amount of time you take to create a potion/scroll. Valet familiars gain the feat even if they do not meet the requirements.
If your crafting limit was 1,000 per day, with a valet familiar, this limit is now 2,000 per day. But what the feat helps with is the Crafting Progress you make each day. If you would take 10 days to craft a 10,000 gp magic sword, with a valet familiar helping you, this task now takes 5 days.
For potions and scrolls, that are normally 250 gp per 2-hours work, it doesn't become 500 gp per 2-hours, but means that your daily progress will be limited at 2,000 gp. Which makes the feat only useful for higher level scrolls or potions crafted using a higher level caster level, as potions are capped at 3rd level.
You cannot craft two items at once, nor you can craft twice the amount of potions. You are still limited by other rules:
Potions and scrolls take 2 hours per 250 gp on their price, or 1 day per 2,000 gp on their price.
Other magic items take 1 day per 2,000 gp on their price.
The GM could house-rule that it also increases the 2-hour limit for potions and scrolls, but that's not what says on the feat.
Best Answer
Ok. This is a bit of a tricky one. Potions of Healing, while created of magical substance aren't actually considered magic items. Or are they.
They aren't magic items
Note, for instance, that they aren't listed as "magic items" in the DM book we've seen so far (BD&D DM Book pp 59-60, v.1). They are listed in the "Adventuring Gear" section of the Players book in basic (BD&D Players Book pp. 48-50 v.2).
In other words, potions of healing aren't magic items in the sense that we think of magic items. They're much, much more common and easily brewed. So following the crafting rules in the PHB should be just fine.
They are magic items
Note for instance they are listed as magic items in the Starter Set DM book (LMOP 53). The entry does say that they are a magical red fluid.
Also of note is the Adventurer's League instructions call them out as a specific magic item that can be purchased by adventurers, not subject to the normal lack of magic item economy in organized play games.
So where does this leave us? In a great deal of confusion on this issue until December. Largely this is because the crafting rules beyond the mundane have been left to DM discretion (with the possibility of further guidance in the DMG). So the answer to this question is, ultimately, if you're a player, discuss it with your DM. If you're a DM, talk to your players and make a ruling. Considering the free and easy access to healing potions, I think it makes sense to allow their crafting, but if you wanted to limit them, that would be a reason to make them hard to craft. Judge the tone of your campaign.
DMG Update
Potion brewing is covered under crafting magic items in the DMG. Under these rules, the potion's rarity determines the potency, but also the amount of time and money it requires to brew. Under these rules, a common healing potion (which normally can be bought for 50 gp), requires 4 days and 100 gp to craft (and cannot be crafted until 3rd level and even then only by a character with spell slots). This is probably not a sensible way to obtain healing potions without a house rule.
Xanathar's Update
Xanathar's Guide provides optional rules for crafting during downtime. In particular, the optional rules (XGE, p 130) under "Crafting an Item" state: