I think the scenario is the 2005 assassination attempt succeeds and Cheney dies of a heart attack upon hearing the news. So there's no time to nominate a veep and Hastert is the next man up.
By the time we find out he's a pedophile he will have nominated a VP and Congress will have approved the pick. Most likely a placeholder/elder statesman (or woman - why not check off a first while you're at it?)
When Hastert leaves in disgrace that person is the new President but has no political base and no chance of winning a full term.
No offence but Jim Jordan, ie he is credibly known since about 2015 to have been involved in similar ‘shenanigans’ and not only hasn’t been driven from office but is rather more powerful than before. Worse, no one seems to talk about it or care.
Perhaps I’ve just become cynical at events 15 years later but I’m not actually confident that Hastert’s involvement in molestation would come out let alone he resign in disgrace. I think it would instead be covered up under the guise that it’s better for the country to not know than to reveal their wartime President was involved. Especially if the powers that be and who have knowledge can convince themselves his involvement was tangential such as providing hush money rather than personal involvemen. I mean as it is the events happened in the 1970s? and he was elected Congressman for the area for a decade and then Speaker of the House for 6+ years without it being spoken of (at least such that local or national media referred to it). If he’s President I’d think there’d be more pressure to keep quiet and it would be even more scary to go up against him or mention it. in this case America is traumatised by losing W and Cheney in the space of a day, wrapped up in paranoia and patriotic fever and facing harder times in Afghanistan and Iraq while staring down Iran, North Korea, Venezuela, Russia and China. It would be an easy response to say ‘it’s [X] propaganda’. After all who would conceive of such people surviving in American politics or the elites at the time (before the GOP scandals of 2006, Weiner / Spitzer / Weinstein / Epstein / Cosby / Trump came to public view).