Scott Aaronson spoke about this in a colloquium where he said that this was mooted at OpenAI before the decision was made by Altman to not implement it for the reasons you describe.

https://youtu.be/9udWn1Hlj_s?si=VWOiK5-y4zcyDoHI

OpenAI will soon be adding watermarking to text as well, as it signed the EU Code of Practice on Transparency of AI-Generated Content: https://openai.com/index/advancing-responsible-ai-across-eur...