The human immune system
Basically, the human (and that includes all ages) immune system has two parts:
- The innate immune system is a very old part (which doesn't mean it's bad or superfluous, on the contrary) that is responsible for a nonspecific immune response when the body encounters a pathogen. This is a very quick response and includes inflammatory responses and fever. Since most bacteria grow better at a temperature of 37°C than at a slightly higher temperature, a fever is a normally beneficial response to any kind of pathogen and thus triggered often
- The adaptive immune system, also called the humoral or specific immune response. Evolutionary speaking, this is newer. The main components are B-cells and T-cells and it is pretty complex, but the basics are that this is what reacts to antigens to produce antibodies to specific pathogens. After the first encounter, this makes sure that the next time the same pathogen is encountered, the immune response is quicker and shorter.
The immune response to vaccinations
The infant's immune system really is capable of handling a lot more antigens than it is exposed to in a vaccine + its live environment at any one point in time. I suggest you read the review by Paul Offit, who is cited in the article you linked to, for more information on that, Addressing parents' concern: Do multiple vaccines overwhelm or weaken the infant's immune system. Basically, the specific immune response mediated by B- and T-cells can handle a specific immune response for a large number of antigens at once.
This number is largely independent of what you are seeing in the child. The number of antigens the immune system can handle says nothing about what symptoms the immune response will cause, because the effects you can see is largely the innate immune response. Small children get a fever pretty often because most pathogens are new to them and the innate immune system responds to that with a fever. Exposure to several pathogens at once does not cause a higher fever.
With vaccines there is no actual risks of the child getting the disease. The vaccines contain attenuated or dead viruses or no pathogens at all (the tetanus one, for example, is against the toxin, not the bacterium producing it). The adaptive immune system needs days to weeks to produce the correct antibodies in sufficient quantities, but it carries no risk of them suddenly developing measles or something like that. Unfortunately, we can't really tell the immune system we only need the antibodies, not the cytokines all the other parts that may let the child appear sick when it really isn't. So the innate immune response happens and may appear to weaken the child and leads to parents to jumping to the conclusion that this is too much for such a small child. Nobody is saying a fever is fun for the infant/baby of course, but in essence, vaccination is a risk/benefits calculation.
One thing to keep in mind is that several of the vaccinations given in childhood are combination vaccines (MMR, DTP,..), and the immune response is still not in any way overwhelmed. And these are several vaccines for diseases that are actually dangerous (diphtheria, pertussis,...) given together. The resources don't get used up because different cells are doing different things and, even more, the B-cells that respond to the measles vaccines are not used for the immune response towards the rhinovirus the child is exposed to at the same time.
The number of antigens is used for that reason - it's a metric to show that the humoral immune response is really capable of doing a lot of stuff at once. It's just that many people see the fever vaccinations can cause and then conclude that a child is actually sick after a vaccination and therefore their immune system must be "weakened" (which, outside of actually immunocompromised people is another ill-defined concept) by the vaccine and incapable of dealing with other things at the same time. It isn't. That fever is the body unspecifically reacting to anything that warrants an immune reaction, because the specific and unspecific parts of it didn't evolve to be independent.
Also, nobody is saying anything about "straining" the immune system, because "straining the immune system" does not have a defined meaning. What they are saying is that a child's immune system is more than equipped to deal with several things at once.
That delaying vaccinations for fear of overwhelming the immune system is unnecessary can be seen by what happens in children who are already sick and receive a vaccine. The answer is "not much out of the ordinary". The innate immune response is launched already and the adaptive immune response can deal with everything at once.
vaccine-specific antibody responses and rates of vaccine-associated adverse reactions of children with mild or moderate illnesses are comparable to those of healthy children. For example, the presence of upper respiratory tract infections, otitis media, fever, skin infections, or diarrhea do not affect the level of protective antibodies induced by immunization
The recommendation to delay vaccines when the child has a serious illness is not based on the child's immune system not being able to "handle" both, but on trying not to "mix" symptoms and avoiding adverse vaccination reactions on top of an illness.
Vaccination should be deferred for persons with a moderate or severe acute illness. This precaution avoids causing diagnostic confusion between manifestations of the underlying illness and possible adverse effects of vaccination or superimposing adverse effects of the vaccine on the underlying illness.
General recommendations on Immunization by the CDC - also a very good read.
The actual study behind that WSJ article is On-time vaccine receipt in the first year does not adversely affect neuropsychological outcomes, though that deals only with neurological outcomes and not infections.
Timely vaccination during infancy has no adverse effect on neuropsychological outcomes 7 to 10 years later.
As a general note on delayed schedules, there has been no evidence presented so far that delaying vaccines has any benefits whatsoever, for example less or less severe infections during childhood. Of course the vaccine schedule is not evidence-based down to the weeks where everything is given - of course parts of it could be delayed by two weeks, other parts could be given two weeks earlier, etc. Nobody is going to design 200 different schedules, enroll thousands of children in a study and test all possible schedules.
But what we have is definitely not overwhelming the immune system and for most children, it ensures that by the time the body encounters the actually infectious pathogens in the wild, the specific immune response to it is already in place and parents will never even notice their child was exposed.
Additional sources and further reading
CDC - Principles of Vaccination
WHO - Vaccine immunology