Why not just pump directly to the $3^\mathrm{rd}$ level directly in a $4$-level laser system? In a 4-level laser system, the pumping is usually from level 1 (the ground state) to level 4 (the highest-energy state).

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Why do such schemes not pump directly to the third level? Will this decrease the output intensity?
 A: Typically,  the 3rd level has a long lifetime compared w/ the 4th level.  For various reasons related to this lifetime, it's harder to pump directly up to that level (long lifetime generally means smaller probability of up-pumping as well as emission).  The 4th level is chosen for its very short lifetime, decaying quickly to the 3rd level.  This lets the user pump the stuffing out of (actually, into :-) ) the 4th level, and indirectly populating the 3rd level very fast.
In the same way, reversed, the 2nd level has a super short lifetime, so the laser transition between 3rd and second levels always "sees" a sparsely populated 2nd level, thus maintaining the strong population inversion.  
A: If you have a laser pumping the 1-3 transition, it will work fine at first, but once the population in 3 starts going up, the pumping efficiency will go down, because so many 3 states are already occupied. Beyond a certain point, pumping further becomes impossible, because my pump laser induces stimulated emission 3->1 just as much as it induces absorption 1->3.
A three-level laser is possible but I think (other things equal) you'd be better off with just levels 2,3,4 than with just 1,2,3 in your diagram.
