Women is accurate!
You think you are being clever and kind and inclusive by using people when you mean women but you aren't.
It is so important to communicate clearly. There are women who won't know this message is aimed at them unless it speaks directly to them as women and doesn't rely on them knowing which "people" have a cervix.
Martin Lewis (Money Saving Expert) said that when they created form letters for one of their earlier consumer campaigns they were taken aback by how many people weren't able to progress to the next stage because they didn't know to or forgot to replace the [your details] sections with their own info. They had taken it for granted that people understood how a form letter worked.
It is so important to genuine inclusivity to be as straightforward as possible.
It's no good sneering at people who don't understand your clever newspeak. Where you say "people" but expect everyone to understand you mean the "people-formally-known-as women". Their healthcare matters too.
If TRAs actually gave a shit about whether transmen were included in smear tests then they wouldn't have campaigned to change their sex marker on their NHS records. Also I would think that the vast majority of transmen know that advertising reminding women to attend their smear tests doesn't have to revolve around their minority gender identity because it's not all about them and they will get over it. As you so wisely pointed out.
"Intersex" people know whether they are men or women with DSDs. You know that right? Even in situations where their sex wasn't immediately obvious at birth, they will know as adults of smear testing age whether they'll have male or female medical care needs.