Fingering is extremely important in the following sense: you should always use the same fingering in the same note.
The reason is very simple. If every time you play a piece you use a different fingering, your unconscious will have a choice to make. This choice will be on the way of flowing, unimpeded playing. Instead all sorts of inaccuracies, hesitations and stuttered playing will result. So not you only you must play the right notes at the right time, but to that you must add, with the right fingers. So as you can see, to me fingering is as important s the right notes (and the right time).
Fingering also implies movement: the best movement will always necessitate a particular fingering configuration. And perhaps most importantly, there is the matter of “preparation”, that is as you play a particular note, your whole playing apparatus should already be “prepared to play the next. You cannot have preparation if fingering is sloppy (that is, if every time you use a different fingering in the same note).
Some people can (usually after years of practice) simply use the correct fingering (for them) consistently without any though to the matter. Fingering has become unconscious, and if this person goes on to teach, typically they will not bother with fingering, and in fact cannot understand why the student cannot do the same. This might explain the case you mentioned.
Now we come to a most important question:
What is the right fingering?And the fact is, the right fingering will depend on the player and on the passage, so one cannot come up with the ultimate fingering. The pianist must find this out by himself. This is the main reason scores often do not have fingerings: They leave it open to the pianist to figure out his own best fingering. On the other hand, scores who do have fingering, should always be regarded as a starting point, a simple suggestion, and no one should ever be compelled to follow such fingerings.
There is one important exception to this rule. Sometimes in technical studies, a fingering is specified that must be followed simply because the aim of the study is to work on that particular fingering, even if some other fingering could be used with greater ease.
How is one to find out the best fingering, that is, what criteria should one follow?
I have discussed this at some length here:
https://pianoforum.net/smf/index.php/topic,2749.msg23873.html#msg23873(reply #6)
And you may find this discussion on Chopin’s fingering for Etude Op. 10 no 9 relevant as well:
https://pianoforum.net/smf/index.php/topic,4957.msg47444.html#msg47444Best wishes,
Bernhard.