I began withdrawing SS at age 62. My wife is retired and receives a state teacher's retirement pension. SS was never withheld from her teacher's compensation. The SS administration says she does not qualify for benefits because she never met the 40 quarters of contributions required, and her pension benefits are too high to allow her to qualify for a spousal benefit. So - are we doing the best we can?
Probably so. Due to the government pension offset provision, Social Security would withhold from her spousal benefits an amount up to 2/3rds of the amount of her teacher's pension. So, if 2/3rds of her teacher's pension is more than 1/2 of your full retirement age benefit, she would not receive any spousal benefits.
The same thing applies to potential widow's benefits if you die before her, however, the widow's rate would be higher than the spousal rate, so she may be eligible for at least some benefits at that point.
One possible bit of good news is that your wife can still qualify for Medicare at age 65 as a spouse on your record. That's the case even if no cash benefits are payable to her.