I post this frequently:
She's beautiful, cool, the Elder God-tier character archetype (kuudere), she's cute, innocent, fun to pick on, loyal, independent, smart/calm, respectable, tragic, kind, dorky, modest, the artist behind her is crazy about her, she's mother fucking King Arthur, she's badass, an empowering female character, charming, charismatic, confident, I could go on if I had all day. Takeuchi was a genius with her design. It should be no wonder why she's so popular. The only people who wouldn't understand why she is as popular as she is haven't read or even seen the Fate route, which has problems, but excels on fleshing out her tragic story and resolving her struggle.
Saber is only "dependent" on Shirou in title. Physically Shirou is the one dependent on Saber, while Saber is sort of dependent on his idealistic views which convince her to except her life and all its unique tragedies, much like he struggles to do. Without each other they wouldn't have been able to conquer the HGW and themselves.
They are my favorite couple because of this.
I think most people also recognize her as "the face of Fate" before they are introduced to her character and so some are hyped up by this, expecting her to be somehow perfect. Saber is only as good as her role in the VN, which is wonderful and endearing. She's just a nice design in all tertiary series, but only truly shines in the VN and specifically in the Fate route, though she's really charming and endearing in HF as well. Being exposed to her in any other route/story first (Zero, FGO, F/A) would leave a bad impression of her because her original character arc was written BEFORE she was as popular as she is now or when those stories were written.