Are you sure that it would be impossible for Godunov to insinuate herself with the new Tsar?
It is impossible to bite your own elbow but in the political issues we are talking about the probabilities.
During Feodor's reign Godunov was not quite friendly or excessively generous to the Nagoy family so they had all reasons and opportunities to indoctrinate Dmitry. Then, of course, there was a political opposition to Godunov which was quite strong and which he could not completely eliminate during his reign. Now, if he is a Prime Minister, how could he completely isolate Tsar Dmitry from all these influences? It is 24x7 work and he has a rather troubled state to rule so he can not be around Dmitry all the time and there are plenty of people with a right of a direct access due to their positions at court. How much would it take to get into the teenager's good graces by making him presents and telling that he has a right to do whatever he wants and not what he is being told to do? As I said, Peter II is a perfect example for this scenario but Peter I would do as well: as soon as Dmitry is of age, for how long is he going to tolerate anybody's dictate?
I mean, while rare, its not entirely unheard off for a favorite/ministers to survive the succession of a new monarch; both the Duke of Buckingham and Cardinal Mazarin survived the deaths of their initial patrons and thrived under the new ruler (though obviously Buckingham didn't have a happy ending). It definitely wouldn't be easy but its not impossible.
Buckingham, whatever was his relation with King James (is it correct that their relations were ...er.... excessively close?), made himself a buddy of Charles when he was a prince and, AFAIK, he did not make himself excessively busy with day to day statesman's duties. A dashing person like him was definitely attractive, just as Ivan Dolgorukov was to Peter II.
Mazarin was not Prime Minister during the reign of Louis XIII, he was allegedly quite close to Queen Regent, he was forced into exile at least once and he had a good grace to die just when Louis XIV came to age.
Or, if Godunov does fall and get exiled/executed, who do you think would succeed him as Chief Minister? Or do we have a slim possibility that Dmitri might take the reigns himself, like Ivan did at his coronation?
Godunov's position during the reign of Feodor was unique. There was no standard equivalent to the "Chief Minister" position within Tsardom's administrative hierarchy.
That's also true. But the book I read on the False Dmitris, This Crown is Mine by Benjamin Levin, suggests that False Dmitri I considered marrying her to consolidate his position in Russia but ultimately decided to honor his betrothal to Marina Mniszech.
Speculations on that subject are endless and it is quite difficult to figure out what really took place or was intended. We can only talk about the resulting facts.
But there's a major difference between Boris Godunov, scion of an ancient, distinguished lineage and brother-in-law of the Tsar, and Menshikov, a jumped up commoner favorite and effective pimp to Peter the Great. Though the Minshikov/Godunov comparison is an accurate one.
Make allowance for the different times: of course Menshikov's ancestry is somewhat murky but what can be said about ancestry of a person he helped to make an empress? She was a washerwoman (which was a common knowledge and which she never denied) who passed through few hands before she ended up in Peter's bed. Different times, different standards.
Who was blind (possibly blinded by Godunov) from 1595 and a not a Rurik dynast. So even at his best Simeon would never have reached the same level of threat as Dmitri, the undisputed son of Ivan the Terrible, brother of Feodor the Pious and last male heir of the main line of a seven-hundred-year-old dynasty.
Unlike Dmitry, Simeon already was a Tsar (
Grand Prince of All Rus), Khan of Qasim, and had all court connections needed and a vow of loyalty to Boris Godunov included explicit denial of loyalty to Simeon and his descendants. So he was quite a serious figure. As for him not being a Rurikid, descendancy from Genghis was practically as honorable as one from Rurik. OTOH, Dmitry's legitimacy could be easily questioned because marriage of his parents never was official. Besides, it is not like with the death of Dmitry the Rurikids would completely disappear or that there was something sacred in being a member of that dynasty to qualify for the Russian throne: Boris was crowned with Shuysky family being still in existence and there were other Rurikids who never pretended to the throne, like Romodanovsky, Putiatin, Gorchakov, Obolensky and numerous others. OTOH, Godunovs and Romanovs were not Rurikids and neither was Prince Wladislaw to whom Russian throne had been offered.