Actually Jade Goody had an even rarer form of cc, adenocarcinoma, usually missed by pap testing, but it's linked to HPV.
The even rarer types of cc, like small cell carcinoma, are also, usually missed by pap testing.
So HPV primary testing and HPV self testing from age 30 is much more likely to catch some of these cancers.
Countries that follow the evidence and make an effort to protect women, exclude them from pap testing and HPV testing until they're 30. Sadly, no country in the world has shown a benefit in terms of incidence or mortality from cc in women under 30, despite cervical screening. Sadly, these very rare cases occur whether you screen or not. Young women tend to get false negatives, which may falsely reassure and lead to a delay seeing a doctor for symptoms. (poorer prognosis) Most of these cancers are diagnosed after the woman becomes symptomatic. (even if she has smears)
But pap testing those under 30, and especially under 25, leads to huge numbers of false positives that can lead to potentially damaging "treatments". Damage to the cervix can lead to premature babies, the need for c-sections and cervical cerclage, miscarriages etc.
The Dutch and Finns have aways protected their young women and excluded them from testing until they're 30. HPV testing should not be offered before age 30 either as about 40% og those under 30 would test HPV+, transient and harmless infections that clear within a year or so, at age 30 about 5% will be HPV+, these are the women who should be offered a pap test.
Following the evidence and targeting those at risk is the best way, IMO, to tackle this fairly rare cancer.
The Dutch and Finns have properly advised their young women too, see a doctor with persistent and unusual symptoms. We tell our young women to have a pap test - falsely reassuring them/misleading them.
Here is Australia we test from about 18 (some even younger) so young women really suffer under our program, it's a highly lucrative source of income though for vested interests, so we'll do HPV testing from age 25 under our new program. You only need to do basic research to find HPV testing is not a good idea before age 30.
Vested interests win again...