I bet she folds when some Muslim soldier’s family wants to “hold a service in Arabic.” In which the imam gets to chant about Allah about 40 or 50 times. They may duly provide her the text — in Arabic of course. And she’ll rubber stamp it.
Should Christians secure the services of someone who speaks New Testament Greek? Jews, Old Testament Hebrew?
This is a key point. If the article is correct, she's prohibiting all forms of religious expression, rather than just some forms. The good thing about that is it's quite possible this will antagonize enough Jews and liberal Christians that veterans will get significant backing from groups like the ACLU which often take the wrong side of these issues, but generally defend free speech rights.
The bad thing is that it's legally much easier to bar all religious expression than to single out some forms as prohibited. A company or a government agency has every right, for example, to tell its employees that they will not use official email to send religiously-oriented messages and will not discuss religion on company/agency time.
I strongly suspect this is “a bridge too far” and will antagonize so many people that the cemetery policy gets changed and this woman loses her job. However, if that doesn't happen, serious damage could happen because the policy may actually be legally defensible on the grounds that failing to bar unsolicited religious statements could imply government endorsement of religion, and could set a horrible precedent if it were to stand up in court. That's not the current state of American law, but a bad precedent could create that outcome.
Let's just be glad this is happening in Texas and not San Francisco or New England where the judges might be less sympathetic.