We seem to have established in the comments that the problem is not with reach, but with reach to. (Secondarily, you should use the indefinite article here, thus a high standard of living or a high quality of life.)
Collins provides various definitions of reach, all based around the idea of arriving at a place or stage, or close enough (literally or metaphorically) to touch; that position is identified by the direct object. Something you reach to is something you are stretching or expanding toward.
The ship reached port with barely half its cargo.
By the time you reach death, you will have walked the equivalent of five trips around the earth.
The old man's beard reaches to his waist.
We reach to the stars because we are bold.
If you think of a society attaining a high standard of living or a high quality of life as a point on a journey, or as a level of achievement, then reach a high standard of living or reach a high qualify of life is perfectly fine, as would many synonyms signifying achievement; a society might attain, realize, or achieve a high standard of living; it might make or manage it if the situation is tenuous, or secure it if you're confidence in the permanence of that state.