Do some not say it on חול המועד?
Yes it is mostly among nusach sfard.
Looked around a bit, mostly seeing people denouncing it.
https://hebrewbooks.org/pdfpager.aspx?req=41190&st=&pgnum=157
Regarding the בוטשאטשער see here, not quite fair to quote him as supportive.

There are Rishonim that explicitly say to say it מוצאי שבת , and it seems the Arizal and בעל שם טוב said it then as well.
Rav Moshe says it isn't considered a legitimate minhag: (he’s talking about the very scenario described in the OP)

Looking specifically if someone gives the reason aygart mentioned.
Finally found someone who gave this reason, see דברי יציב סוף סימן ק״ג,
https://hebrewbooks.org/pdfpager.aspx?req=15079&st=&pgnum=206although he doesn't seem be a fan of the minhag. Interestingly though he says that wouldn't work for חול המועד because work for דבר האבד is permitted. However as referenced above we do find that reason given not to say במה מדליקין.