In some of your examples,"get to do" can idiomatically mean that you were allowed, or enabled to do something you are happy about doing. I think that is what you mean by "willingness". For example:
"I got to stay in bed this morning" means that circumstances allowed you to stay in bed, and implies that you are happy about that (you must be, otherwise you'd get up!)
"I get to finish work early" suggests that you have been allowed to finish work early and that you are happy about that (who wouldn't be?!)
To be clear though, this is idiomatic of that entire phrase "get to do" and in the right context. Not all of your examples are like that:
I got to the train on time.
Although you would almost certainly want to catch your train on time, this really means that you reached the train on time, not that you were "enabled" or "allowed" to. Also, "getting a train" means the same as "catching" the train. Neither of these contexts are about what you were able to do but are about what you did. For contrast:
"I get to go on the train" means you have an opportunity to go on the train and implies some willingness.
"I get the train" means that you catch, or get on the train. It says nothing about whether you enjoy catching it or not.
I should add that it is common to use "get to..." in a sarcastic tone, which would, of course, imply the opposite - that you are not happy about doing something, for example:
She gets to go out while I get to babysit.
In a certain tone, I would interpret the above as someone being bitter about being the one that has to stay home while someone else goes out. Context and tone are always important.