However, Saurus really only compete in the Close Combat Phase (ok, the Movement Phase, too, if you kit them out accordingly).
That's two phases, and compared to Ogre Khan, that's pretty similar, except we pay a lot less for more wounds and attacks and armour - Ogres rely entirely on the charge, which we don't, and getting a few stomps, which doesn't work against everything.
Saurian Warriors get slaughtered far too easily, and need serious buffs just to equal other combat units that are significantly cheaper.
As has always been the case, they are balanced around magic support, same as undead, which is not a benefit. I'll admit that is a huge problem, and considering the love and attention zero - magic lists are getting (banner and cheaper dispel scroll) it would be nice getting a few benefits for our Saurus if we decided to make a non-magical list.
every time I bring a serious unit of Saurian Warriors.
I think this is your problem. Saurus don't work as a 30 man combat unit. Split that unit up in 3x10. That is a lot more map coverage, advantage in the deployment phase, and a lot easier getting the fights you want, and breaking those steadfast units.
Saurus are good enough as a core unit. They can soundly pummel the snot out of some of the other races' core, but in turn gets beaten by others. However, the "serious" combat blocks of Saurus will never fare well, because any similarly "serious" unit will never be core, it will be an actual elite close combat unit.
The main problem we have, is that we CAN'T field an actual elite combat unit. Temple Guards just doesn't cut it, being just a minor variant of Saurus warriors. For an army with very limited range, we are very bad in the Close Combat Phase for some reason. This is really the one thing I hoped 9th age would have fixed.