The Zohar states:
דודאי לא ברא קודשא בריך הוא מלתא דלאו איהו צריך
It's certain that The Holy One blessed be He didn't make anything He
Everyone is needed; everything has an infinitely significant purpose.
Note the subtle point here. If a person was created for their own need, that's circular1, arbitrary2, and pathetic; it means their life is ultimately meaningless, and they are needy - a bit like a pet, ch'v. This is the root of much if not all neuroticism. See Chassidic Discourses Vol 1:18. for an in depth discussion of why a human is an intrinsically purposeful creature.
This purpose breathes life and vitality into everything else, otherwise it would all just be הבל הבלים, vanity. The pursuit of happiness, pleasure, satisfaction, self determination and mastery of one's life will fall hollow and empty in the end, without it.
It is infinitely better, morally and mentally, to be needed, vs being needy.
Note that we all have a general purpose (fulfilment of the Mitzvot), as well as more specific ones as well, such as our tikun, our yiud etc. Statements of "you were created to open that one door for that one lady" do not imply you are not intrinsically necessary in and of yourself. As Rambam says in Moreh Nevuchim 3:13:
שלא באר כלל בדבר מהם שיהיה בעבור דבר אחר, אלא כל חלק וחלק מחלקי העולם
– זכר שהוא המציאו ושמציאותו היה נאות לכונה
For no part of the creation is described as being in existence for the
sake of another part, but each part is declared to be the product of
God’s will, and to satisfy by its existence the intention [of the
See ספר השיחות - ה׳תנש״א on כי תבוא for an exquisite explanation of how Hashem desires us, ourselves, rather than as a means to an end.
So, statements like the one you've brought about the old lady need to be given their proper context, as they can't mean that you are otherwise unnecessary - you are. So what they really mean is that, even if you waste your whole life, except that one small act of kindness, it still was worth it for you to be born because it was significant, and only you could have done it. Worth it for you, for her, for all of Klal Yisrael, and above all: for Him.
1 - Pirkei Avot 4:22: "you didn't choose to be made"
2 - See this answer for discussion on the evil of arbitrariness