I'm not sure there's any need to have some spirit of the hero in order to pull out the Master Sword. The Hero of Winds, allegedly, was not connected to previous wielders in any way, but was able to pull the sword after jumping through some hoops and combing his hair in a particular way. In Wind Waker, the closest analogue is probably the Mjolnir of Marvel Comics that puts more emphasis on personal worth and valor than it does anything biological.
That is until Breath of the Wild, which changed how I perceived the Master Sword.
We know that Fi faded into the Master Sword at the end of Skyward Sword and that she speaks to Zelda several times during the events of Breath of the Wild. Seeing as she is self-aware in the sword, I don't think it's unreasonable that Fi is actually the one determining worthiness whenever anyone tries to wield the Master Sword.
And then BotW Link fought Ganon and lost. Not only did he lose, but he was beaten to near death and would not have survived without the intervention of his Princess Zelda.
This is the Master Sword following its disastrous encounter with Ganon:
The blade has been physically mauled. Chunks of it are
missing. It took a hundred years for the Master Sword to piece itself back together and even then it's barely functioning. It operates at half power most of the time and is capable of operating at full power for only brief periods of time.
It tries to kill Link.
This can be justified however you want by game mechanics, but the Master Sword in Breath of the Wild actively tries to kill Link when he draws it. I've seen other players suggest that this is some test of worthiness, but this is the
only time we've seen the Master Sword lash out with any sort of fatal intent.
It's also the only time the Master Sword was physically damaged in such a way that its functions were crippled.
Actual theorizing time:
Fi has always been in the Master Sword and is solely responsible for dictating who and who may not wield the blade and who is or is not worthy. Unfortunately, this effectively made the Master Sword her body when she gave up her physical form to inhabit the blade. And then the Calamity tore the hell out of Link and savaged the Master Sword until it could barely function--essentially doing the same to Fi.
So Fi shuts down. She goes into low power mode, turns on all the alarms, and gives the Master Sword time to reconstitute itself. She's near the end of the process (which I'm assuming based on the Master Sword regaining its full power immediately upon completing the Master Trials) and then some asshole comes into the Lost Woods and tries to pull her out just as she's finishing up repairs.
Automatic defense systems go up: the Master Sword tries to kill Link to prevent him from interrupting repairs. Depending on how many heart containers Link has, this succeeds or fails.
Long theory short: Fi is the one determining worthiness to wield the Master Sword and said conditions are arbitrary.