Why should I build SHIVs?
Solution 1:
The real benefit from the SHIV that you'd get in normal play (sans Enemy Within) would be a mechanized unit you can bring into battle with you that doesn't need training and you don't need to worry about getting destroyed (other then money costs, of course). If you're like me, you never want to see your squad members die in combat. You spend a ton of time leveling them up. So when it happens, it feels like you've wasted a few in-game hours by getting one of them killed.
A SHIV, on the other hand, doesn't need that type of investment. And I find it's main role comes in what I can't do with my squad members without getting them killed: revealing the map. Since your squad members are so valuable, you might be like me and only inch them out a few steps (this goes along with MEC units in Enemy Within, simply because of how costly those are to train up). With a SHIV, if you're so concerned and want to reveal an area without getting your assault blasted, you can send in the SHIV. Sure, you might lose the thing, but since the basic one only costs you around $100 with 0 engineer discount, you could crank out a few and send them out.
And since the Alloy SHIV's ability to create cover gives you both a mobile defensive position AND something else to shoot at aliens with, it's a viable options.
Granted, I don't think I would use them after I've leveled up a good MEC unit, since they can have pretty high mobility with the Kinetic Strike and they're beefy as well, but it's something cheap that doesn't cost you a good squad member.
Solution 2:
Are you kidding? SHIVs are incredible in Enemy Within, especially in the end-game. The new research options for SHIVs not only give it a power identical to "Close Combat specialist" (take a free shot when an enemy comes within four squares) but ALSO the Sentinel Drone, which can heal the Shiv by 2 HP each round... for the ENTIRE mission.
In other words, you know those terribly long missions that are a grind, where you constantly have to watch HP? SHIVs make it a snap. Just send them in as the front line, pull them back once they take some damage, give them a few turns to heal back up to full health, then continue.
As if that weren't enough, SHIVs are totally unaffected by Psionics, unlike MECs (who are just slightly more resistant). You know those annoying Sectoid Commanders who are always in the control room of ships? You can take them out easily by sending one SHIV into the control room all by itself, leaving your soldiers out of sight so that they don't get mind-controlled to attack the Shiv. The only thing that the sectoid commanders will be able to do against a Shiv is use their pitiful plasma pistols, which do even less damage against a SHIV (because SHIVs have the Hardened trait).
Solution 3:
If I just want to get done with missions quickly, I would use all shivs. In late game, I want to collect tons of money and finish all research. Unfortunately, occasionally, I got "missions". Otherwise I'll just get money after money after money.
So what?
I can finish missions much faster.
I can send alloy SHIVS or hovering shivs to rush enemies. No room can survive that.
Anything goes wrong, I just "rest" by pressing back button. The SHIVs auto heal.
The only downside is I don't get rank.
SHIV are like heavies or assault.