Your analysis is fine. It is grammatically incorrect. It should be "Rester chez soi" (infinitive), or "Restez chez vous".

rester chez soi is grammatically correct in my opinion. As an example you would say « stay at your own place is better than staying at your mother’s place ». Rester chez soi est mieux que chez sa mère
But hey, this is one man opinion

It’s technically grammatically incorrect but I think that’s failing to see the forest for the trees. This tweet is part of a cross-border campaign and, as the whole context shows, the French phrase is apparently lifted from the video and the subtitles in the Belgian prime minister’s contribution rather than something the Dutch prime minister wrote out of the blue.

In this context, the most striking thing linguistically is that “rester chez soi” is not a good translation for “blijf thuis” or “bleib zuhause”. If it was, the “z” would only be a small typo or transcription error. As others have pointed out, “rester” and “restez” sound exactly the same in French so the spelling mistake doesn’t make the whole sentence feel agrammatical.

In fact, the closest French translation would be “reste à la maison”. A less literal but possibly more idiomatic translation would be “ne sortez pas”. Incidentally, both of these refer to one’s home (as opposed to the whole/country region as “thuis” or “zuhause” might in Dutch or German).

If you consider that officials typically use a more formal register in France than in the Netherlands (not necessarily true when comparing Belgium and Germany), it could be “restez chez vous” (German: “bleiben Sie zuhause”, which is what Laschet — but not De Croo or Rutte — says in the video). That’s also the plural form in French (German: “bleibt zuhause”). This discrepancy in tone and meaning is the main reason why that phrase feels improper and presumably hasn’t been produced by a native French speaker.