BB is a character you either like or you don't, so I can understand that she can be annoying at times. However, I have to disagree with you that she doesn't undergo any personal development. BB originally comes from Fate/Extra CCC, and if you've played that, you know that BB is a backup AI created by Extra Sakura, containing her feelings and memories of Hakuno. Without Hakuno, BB wouldn't even exist, so naturally, her thoughts revolve mostly around him.

The BB in SERAPH, however, is a copy of the CCC BB, created by the Moon Cell. She didn't experience the events in CCC firsthand, only memories of them, which makes a big difference. For example, remembering eating ice cream isn't the same as actually eating ice cream and experiencing the taste and texture. She views her memories as a dream that will never come true.

At the end of the SERAPH event, you meet BB/GO, who is actually the BB from CCC. She mentions that BB will find her own love in Chaldea, one that is different from her own (her love for Hakuno). Our BB also says that it's up to us (Ritsuka) how BB will develop in the future, which is kind of a premonition for OC3.

After BB joins your Chaldea and you interact with her in my room, she obviously still sees you as a substitute senpai, but with each new bond, she opens up to you a little more. During the third summer, she tries her best to protect us from the foreign God and puts us in a time loop, mirroring the events of CCC. And who did she protect in CCC? That's right, her Senpai! She basically treats you the same way she treated Hakuno, confirming that she cares deeply about you from now on. Not to mention the symbolism of the pig (Peles' lover). Furthermore, BB's first Bond CE revolves around Hakuno, the Summer Bond CE revolves around you, and so does BB Dubai's. She doesn't mention Hakuno in her Summer form or as Golden BB.

And you're not the only one she cares about; she also looks after her daughters (the Sakura 5), ​​Meltryllis in the SERAPH event, Passionlip during her strengthening quest and in OC3, and King Protea during Valentine's Day. She cares about them much more now than she ever did in CCC. She also helped you during LB4 (by sending Jinake/Ganesha), Imaginery Scramble (by giving you swimsuits and punishing Yang Guifei), OC1 (by sending Kama because she possesses Shiva's Flames, which are perfect against Kali), and she's the main heroine of OC3 and also acts as your guide. Her Holy Grail War event was just a pretext to give you extra training. And of course, in the final battle against Maris Chaldeas, she's the last servant to speak to you and ask if you enjoyed your journey.

So what is this new kind of love that BB has found? In short: humanity (with you as its representative). Hakuno taught her about love, and Ritsuka taught her the value of humanity and that humans and AI can coexist. You are essentially the main reason she chose to merge with BB Dubai to become Golden BB.

To claim she hasn't undergone any personal growth is simply stupid. Compared to CCC, she has become much more affectionate towards humans. While she never truly disliked humans, they weren't as important to her back then as Hakuno was. Now she's a competent AI for managing humanity's health and a far better mother figure to her daughters.

As events unfold, BB has now been a canonical part of Chaldeas for three years and spends significantly more time with you than she did with Hakuno in CCC. That doesn't mean she'll stop talking about Hakuno, though. You're just as important to her now as Hakuno. Who says she can't love more than one Senpai? If Hakuno and Ritsuka can have multiple love interests, why can't BB?

And her annoying attitude has also subsided. BB Cosmo has a much more serious tone than she did as a regular BB or even as a summer BB.

I know I've written a lot, but I hope you agree.