Most real famous people gain their fame through their work or accomplishments (movie star, professional athlete, etc.), becoming famous is just a by-product of that and many people that do become famous often comment how they miss the loss of privacy and anonymity.
Beth just wants to be famous for the sake of being famous, I don't quite get what the desire in that may be, is it just an ego thing? or a poor self-esteem thing?
She is already rich by extension of Howard, why bother trying to gain fame? I think I would rather be wealthy and anonymous than people bothering me all the time or being followed around by paparazzi.
She has invested a lot of time in trying to become famous I suppose, she seems to find every single random event that nobody has even heard of just to have her photo taken, pictures of which will never be seen anywhere other than Dawsaloon. I think the best event she ever attended was that car dealership opening.
Howard on some level must feel embarrassed and sad for her.