NO, SSA does not have any responsibility of notifying spouses when they are eligible. Families are all different in their makeup, ages and dependency status. The SSA asks about the makeup of the family when application is made to the SSDI program and if approved they will give an estimate of which family members may have some part of the benefit.
SSA: Disability Benefits - Family Benefits
SSA: Disability Benefits Publication
You gave no details about yourself - i.e. your age, your current work status, are there any dependent children - so can't really say if this is a good or not good move for you.
Remember that when you apply for this SSDI spousal benefit, you are declaring yourself as retired and you are deemed to be filing for your retirement - your benefit will be either 50% of your disabled spouse's benefit OR your own benefit or some combination to equal 50% of your spouses benefit; whatever is the higher. You have to be at least age 62 to apply or any age if you are caring for a child of the beneficiary who is younger than age 16 or disabled.
If you are working and are going to continue to work, and you have not reached YOUR full retirement age, your benefit will be reduced if you go over the income earnings limit until you reach your full retirement age. (From the AARP link below): In 2021, the earnings limit for early claimants is $18,960. (The figure is adjusted annually based on national changes in average wages.) You lose $1 in benefits for every $2 in earnings above that amount.
AARP: What is Social Security’s 'special earnings limit rule'?
The only thing different about the SSDI program vs the SSOA (old age program) is that the main beneficiary (the SSDI beneficiary) can be of any age because they are already getting their full benefit.
It's Always Something . . . . Roseanna Roseannadanna