Choosing k3 only takes into account the dissociation of the weakest acid (not including water; HPO4- is almost as weak as water) and does not take into account the dissociation of the strongest acid, H3PO4.
Let's do this quantatively.
I don't have the actual Ka's, so I'm going to go off 10^-3, 10^-8 and 10^-13 as I remember those as the correct order of magnitudes.
Let's assume we have a 1.0 M solution of H3PO4.
Then, by our weak acid assumption, Ka=10^-3 = [H+]^2/1
[H+]=10^-3 =>[H+]=roughly 0.032
Technically this means our assumption is getting close to dodgy, but let's disregard that.
Then, let's do the same assumption for the second dissociation. We have 0.032 M H2PO4-
So, using Ka=10^-8 this time, Ka=10^-8 = (0.032+[HPO4-])[HPO4-]/0.032 M
Note that we can't assume [H+]=[HPO4-] because we already start off with a lot of H+ in comparison to how much HPO4- we'll get. Just look at the Ka of 10^-8
But, we can assume that 0.032 is much greater than [HPO4-], so we have 10^-8=0.032*[HPO4-]/0.032
I think we can see that the concentration of HPO4- is going to be around 10^-8, and thus [H+]=0.032+[HPO4-]=still around 0.032
You can probably see, by now, that the second dissociation isn't going to count much. Similarly, the third dissociation will be even more negiglible.
Generally, for polyprotic weak acids, only the first dissociation really impacts the pH.