Not quite accurate. SSA administers two kinds of disability programs. One is SSDI. Anyone (under full retirement age) who has enough recent work credits under Social Security and who establishes that he or she cannot engage in substantial gainful activity, will get benefits. Those benefits are NOT means tested.
The other program is SSI. The disability standard is same, but there is no requirement for a certain number of recent credits under Social Security. SSI benefits generally are much lower and there definitely is a welfare component to SSI benefits. I doubt if anyone on SSI gets $14,000 a year.
Some relatively low wage earners might qualify for both SSDI and partial SSI. I doubt that is your particular situation.