I can comment on this, as a Mormon, and I will say that the Psalms 127 is a very popular scripture, both in the LDS church and out of the church because it famously talks about "Children are an heritage to the Lord." I hear this quoted multiple times per year at LDS church talks because for a religion where the family is the most important and central thing in their lives, it is so relevant. Thus, when the Psalm uses the metaphor, comparing a Quiver of arrows to a "Quiver Full of them," referring to the children, there is nothing at all saying God wants them to not consider how to get pregnant. He in fact is saying to have LOTS of children!
The "quiverfull" movement that you linked is more of a modern movement that was popularized (though not really popular) long after this verse had been around. Mitt Romney probably grew up hearing Psalm 127 and I don't think him quoting this verse shows any support from him towards the Quiverfull movement, nor does the LDS church. Unfortunately, their movement decided to hijack a very popular scripture and now many people like to unfairly clump those that use that quote that scripture with that movement.
In regards to people saying it is somehow inconsistent to be anti-abortion yet pro-IVF, I think that is intellectually dishonest. The general view among people that are anti-abortion is the idea that it is actually murdering the child. Not saying you have to agree, but I am trying to give you a different perspective as to the why, from a Mormon perspective, the idea behind things like IVF and so on are that God gave man intelligence and a brain and surgeries to be used for the good of mankind. Surgery is unnatural with the logic of the Quiverfull movement. Unnatural does not mean against the will of God. Not having a problem with IVF but being against abortion is not hypocritical and frankly, kind of ridiculous that people are trying to push that idea without actually thinking about it.
Again, I'm not downvoting anyone here, just giving some perspective here since you brought up a hint that Mormons might actually believe that ridiculousness, imo. Don't have to agree with it.