Psalm 127 – Skeptic's Annotated Bible answered

The psalmist doesn't say there shouldn't be builders or watchmen. But he says that their labor is in vain if it is not blessed by the Lord. What if one builds a house, but one does not receive the time to enjoy it? What if a watchman wakes and sounds the alarm if the enemy approaches, but the city is still taken?
So indeed, people build houses, but it is God who has provided an earth to do so and health and wisdom to engage in such activities and prolongs our lives to enjoy the fruits of our labors.

The interpretation of the author of the SAB is in direct contradiction with many other parts of the Bible. It is likely that Solomon wrote this psalm. And in another place Solomon says:

Yet a little sleep, a little slumber, a little folding of the hands to sleep:

So shall thy poverty come as one that travelleth, and thy want as an armed man.

And Solomon would say here the opposite? The meaning actually is that God gives sleep, a rest based on a good and still conscience. While others, having been ruthless and merciless will turn on their beds. As John Gill explains:

he gives not only bread to eat, but sleep, which to a laboring man is sweet; and having food and raiment, he gives them contentment, quietness, and satisfaction of mind, which is the greatest blessing of all.

The author of the SAB links to an American movement who uses this text to argue humans have no responsibility with regards to how many children a woman has. God's foreknowledge and governance does not exclude our human responsibility. For example in chapter 91:11-12 we read of God's protection against those we fear him. Does that mean we can put ourselves in danger and not care, because God will protect us anyway? Of course not (Matthew 4:6). Even the opposite. Someone is liable if he does not take reasonable precaution against someone coming to harm in his house, Deut. 22:8.
Do we not need to work because God cares for us, Luke 12:29? Of course we need to work, 2 Thess. 3:10.
And so families also have their responsibility with regards to their children and their health. Having as many children as possible could have adverse health effects and we shall not kill. Some women can easily bear many children, and some can't. There is an individual responsibility here that each family has before God. There is even a specific responsibility that husbands have towards their wives, 1 Pet. 3:7:

Likewise, ye husbands, dwell with [your wives] according to knowledge, giving honor unto the wife, as unto the weaker vessel

Dwell with her with knowledge scripture says.
A woman is also not always fertile. That already indicates that there are times and season and natural fertility periods that can be used. What is clear, and I agree with the movement there, is that chemical means, pills, should not be used. They may have abortive effects, and should therefore be avoided. And even if they have not, it simply cannot be good to take chemicals to regulate one's body in such a crude way. Bodies are not simple and simply too little is known about biochemistry and long term effects to have any trust in such devices.