The point is that luminous intensity is intensity as perceived by the human eye, and particularly taking into account the fact that the same amount of power will be perceived as brighter or dimmer depending on whether the wavelength is at a maximum of the eye's sensibility or at a minimum.
This makes the candela ever so slightly washier than the other six SI base units, because you need to rely on physiological measurements of the average human, whoever that is, but the simple fact is that you cannot measure how bright a light looks to a human eye using just stopwatches, yardsticks and weights - you need to use the eye itself as a measurement device, or at the very least calibrate with one. Luminous intensity is a measure of the biological response of a specific system (if not of the subjective perception this response causes), and unless you're willing to define the unit of luminous intensity in terms of neuron activity on the optical nerve then you really do need a unit of measurement that's independent of the mks triplet.
In general, suppose your eye receives light from a source with a radiant intensity of
$$I_\mathrm{r.i.}=(I_\mathrm{r.i.})\:\mathrm{W\:sr^{-1}m^{-2}},$$
i.e. each unit area $A$, channelling a pencil of radiation of solid angle $\Omega$ receives $I_\mathrm{r.i.}A\Omega$ joules of radiated energy per second. This still doesn't tell you how bright the source will look to you, though, because the different receptors are more or less sensitive to radiation at different wavelengths. However, this dependence has been quite thoroughly studied using a number of methods, which have resulted in a fairly standard luminosity function: that is, a function
$$\bar{y}(\lambda),\text{ also denoted }V(\lambda),$$
that tells you how bright lights look at different wavelengths. Thus if $\bar{y}(\lambda_1)/\bar{y}(\lambda_2)=2$ then a light at $\lambda_1$ will look twice as bright as a lamp of the same radiant intensity at $\lambda_2$.
There is of course some individual variation, as well as the tough metrological problem of measuring and standardizing the luminosity function, and controlling for the fact that different populations can have different average responses, but that's all in the game and it's ultimately someone else's (i.e. not a physicist's) business. And if you want to really go down the rabbit hole, you need to control for the fact that the perceived intensity will vary under well-lit versus low-light conditions, and down and down it goes. However, you only need to normalize once for each set of conditions; hence the value of standard candles.
The candela comes in, essentially, as the units of the standard luminosity function. A light source of wavelength $\lambda$ and radiant intensity $I_\mathrm{r.i.}$ will have a (perceived) luminous intensity
$$I_\mathrm{l.i.}=\bar{y}(\lambda)I_\mathrm{r.i.},$$
where now the luminous intensity $I_\mathrm{l.i.}$ is a completely different beast to the radiant intensity, depending as it does on a human reaction, so it is measured in candela. It follows, then, that the luminosity function has units of $\mathrm{cd}/(\mathrm{W\:sr^{-1}m^{-2}})$, and the role of the SI definition is to normalize it such that
$$\bar{y}(555\:\mathrm{nm})=\frac{1\:\mathrm{cd}}{\mathrm{W\:sr^{-1}m^{-2}}}.$$
From here one can then fill out the rest of the curve for $\bar{y}(\lambda)$ using only comparative measurements of (perceived) luminous intensity, which are much easier.
When measuring the "brightness" of a light source, there is a huge number of different quantities of interest, each with their own unit but a very similar name to the others, and depending on exact details like whether you're integrating over angle, or surface, or wavelength, or any combination thereof. It is very easy, as a physicist, to simply give up and reckon that "luminous intensity" will simply be one of the list. Similarly, it's easy to simply slide over terms like photometry and not realize that it's very different to radiometry.
That just means, though, that we need to up our game a bit and realize that there's an extra dimension at play here - the subjective sensation of brightness as perceived by the human eye as a measuring device - that we need to include on an equal footing to our clocks, meter sticks and (soon to be) watt balances, if we really want to produce measurements which are useful in a world inhabited by humans.
