I don't think TM should be made to resign unless the landing cards were destroyed at her instigation, and it now appears that they were disposed of in 2009, under the Labour govt, not in 2010 under the Conservatives.
I do think though that this whole debate has got out of hand. It's an excellent idea to put the responsibility for providing paperwork onto the various agencies of govt - NHS, local councils, HMRC, etc - who surely have health, housing and tax records for the vast majority of these people. No one can live in a country for 40, 50 or 60 years without having contact with these agencies.
However, I do think that blaming the govt for everything is ridiculous. As a person living in any country you are responsible for ensuring that your paperwork - whatever it is - is in order. If you are living in a country other than the one you were born in this is particularly important. An article in the Guardian today {www.theguardian.com/uk-news/2018/apr/21/stuck-jamaica-pension-house-kids-uk-windrush-britain) quoted a guy from the Windrush generation, who said:
“I did my papers straightaway in 1972 as soon as you could after 10 years but some people ignored it."
And he's right - after 10 years you have a right to ILR and you should claim it. But blaming the govt because you never bothered to regularise your status is naive and dangerously passive. Laws change all the time. Immigration laws, in particular, change regularly and the conditions never become less rigorous, only more. So don't blame others because you never bothered to claim the rights you were entitled to. Sort out your paperwork in a timely manner and make sure that your rights are being acknowledged.