Yes, that's correct. It seems to be that way to me.
In the first example you gave,
When I was on the stage the next day, we can't possibly be dealing with 'the next day' meaning 'tomorrow' because of the 'was'; this event clearly happened in the past.
Your question was why does 'day' need 'the', but month and year don't. My answer now is that year an month need 'the' if they are used in this same way; to mean 'any year that followed the one previously specified', and not 'the next year from this one'. I hope the examples I gave show this:
I was in a play for my school production. The next year I was acting on Broadway.
We will be getting married in June. The next month we'll spend on a luxury cruise as our honeymoon.
As ymnisky said, in speech, people may drop these articles or say them so fast and unstressed that they seem non-existant, but they are necessary in writing. Saying just 'next year' and then 'I was' isn't possible; 'next year' can only be in the future. 'The next year' means 'the year that was next to (i.e. 'after') the year I was in a play for my school production'.
In the same way, 'the next day' means 'the day that was next to (after) the day specified previously'. 'Next day' doesn't exist in spoken/written language; we only use 'tomorrow'. However, if it did exist, it would mean 'the day next to this one', and like 'next month' and 'next year' it couldnt be used in past. The same way 'tomorrow' cannot be used in past tense.
If it's still not clear I'll try and help make it more clear.