Yes, they would have to have more than two breasts.
For one thing, breast-milk is not homogeneous. Colostrum is the very first milk, very easy to digest and specific to newborn infants. Then, as the child matures, milk develops into two parts - fore-milk, the first milk from a full breast, is thin and watery with vitamins and proteins. And also hind-milk, the last milk from a near-empty breast, which is thick and creamy and heavy on fats. In short, the composition of the milk changes over a feeding session, and over a child's growth, likely to prioritize which nutrients are available and address which are needed. Nor are these necessarily discrete substances, it is much more likely a gradation for production and letdown of one to another depending on the child(ren)'s needs - making it harder to equally subdivide each type of milk when feeding multiple infants.
This makes scheduling or rotating feedings much more complicated. Multiple babies rotating between two breasts will likely get differing levels of nutrition, differing amounts of calories, even different volumes, depending the order each eats in a feeding session. And it would be easy for some children to get left behind, undernourished, if there's any problem feeding or if the scheduling or rotating is even slightly off balance - those eating first getting nutrients so the second feeders are malnourished, those eating later getting the bulk of fats and calories so those eating first are starved. On the other hand, each breast has its own supply of each kind of milk (one reason switching breasts when feeding is a thing), so having at least one breast per child should help to minimize that kind of unequal division - though more would be useful backup for when the children need more than one breast's worth of milk.
For another thing, infants eat every two or three hours, can take as long as an hour for a single feeding in some circumstances - and can have marathon feedings, preceding a growth spurt, with feedings every 20 to 50 minutes. Rotating feedings would mean spending twice as much time on feeding, not to mention the extra amount of time it would take to completely refill the breast (to make sure both kinds of milk are produced). And with all the other demands on the parent's time doubled, well, the first few months are already commonly held to be a time of sleep deprivation and frantic activity (cleaning, soothing, keeping warm and dry, self-care), needing to double the frequency or duration of feedings is likely not going to help at all.
An intelligent animal can manage, I believe, as you said (as we do, when multiple kids are born) - but it would be a real and nontrivial effort and require a great deal of support from others. And those others would likely have their own to tend, as this is a normal and not extraordinary circumstance, limiting the amount of assistance each could reasonably expect to get. Also, time may be somewhat less important for having safe and organized villages as opposed to being vulnerable and alone, but it is still quite important - especially since more time per litter would be needed generally for childcare, and more calories would be needed to produce milk (thus more time to forage, prepare food, eat, rest, etc).
It is possible to balance out the feedings so each gets their fair share, but it is a great deal of work - and it is equally possible to miscalculate, to under-nourish one or more of the litter and those so neglected will be much less likely to survive. I would expect that such balancing would be much, much harder further back in evolutionary history - with the effect that either those with smaller litters would out-compete those with more, as their kids are healthier, or else more breasts would evolve.
A third nipple is a possible mutation already, if it gave a concrete advantage I would expect it to spread broadly and possibly eventually a mutation of a fourth nipple may appear, or more, until the number meets or exceeds expected children per litter (to ensure each has a chance at equal nourishment). And each breast may be smaller, or some or all of them may be dormant, flat and not engorged, when not actively breastfeeding a litter. This happens in other animals, so it seems a likely solution if extra breasts would negatively impact the center of balance.
