State workers have it rough. Yea, they have more job security than many private employees. But, if they are not paid in accordance with the Fair Labor Standards Act, they cannot file suit against their employer. That was the result of the 1999 decision in Alden v. Maine. They also cannot sue their employer under the Americans with Disabilities Act or the Age Discrimination in Employment Act.
The Texas equivalent of the ADA and the ADEA provide comparable protections. But, the state version of the Fair Labor Standards Act has no teeth. Most states have a law comparable to the FLSA. A handful of states, including Texas, do not. So, at least regarding wage violations, state workers have little or no recourse.
The 1999 Alden decision was a 5-4 decision by the US Supreme Court. That decision overturned decades of earlier decisions by lower courts. So, yes, presidential appointments to the Supreme Court do matter.