Most people here target retirement before any child who would qualify for this benefit would be born, I doubt you'll get a response from someone with experience in this particular area. Also note that the benefit belongs to the child, not the parent. Of course the parent may spend the benefit for the needs of the child. If any of the benefit is saved, it is transferred to the child when the child reaches age 18.
Personal benefit at age 70 is about 77% higher than personal benefit at age 62. I've seen the numbers run for someone looking to maximize their sustainable spending rate as always being better off delaying SS until age 70. Does the extra benefit received for the children change this? I suppose that would depend on the ages of the children involved.
Suppose the primary insurance amount (PIA) is $1000 and maximum family benefit is $1500. You decided to take a reduced benefit of $700 to retire at 62. The child's benefit is up to 50% of PIA, so $500 in this example. The maximum family benefit would apply while at least two children qualify. I'll assume you collect $1500 for 4 years, $1200 ($700 for yourself and $500 for one qualifying child) for 2 years, and $700 for 2 years between age 62 and age 70. Assuming real, after tax investment returns of 5% at the your 70th birthday then net value of the benefits the family had received to date would be about $150,000. Using the 4% rule, this doesn't quite replace the benefit reduction taken to retire at age 62.
I doubt that there are many families that can actually retire earlier using this a strategy.