I don't know that internalized homophobia is the best way to describe Saki's issues. Her dim view of both herself and her relationships to other people/thoughts on how others view her extends to all of her relationships, irrespective of whether her sexuality is relevant to the situation.
It is since it's the primary source for all of her issues so far in the manga. It ties to her biggest fear that she will be rejected by society at large as well as people close to her if she ever admits to everyone that she is attracted to girls. It is the only issue that caused her to go into full panic mode, especially when Rinne met her and pressed her about what she is hiding.
But, to be fair, that is the only thing manga has shown us so far as well as Saki herself when monologuing about her past. When you check all the scenes from her past the only exception to her sexuality was when she was a kid and had to go outside and play/wait while her mother took care of her grandfather and when she asked her mom about her dad and mistook that she angered her. And only asking her mom had some effect on her, but not too much as she was able to ask other people about sensitive things in the manga and not have panic attack of some side-effect of that.
Not having a dad can have it thought but ih her case it had no lasting effects. Her mom raised her well and she had no issues establishing normal relations with people when she was younger nor now in high school. Nor she was lacking for money as she was taking piano lesions from early stages of her life, and those cost a fortune.
When you look objectively at her as a character and not take the internal homophobia into account, Saki is competent character that is a bit tame but when push comes to show she can pull more than her weight. She is great at interactions with people (especially with children, all kids loved her so far), her studies are solid and she can help others learn as well, she is great organizer and can manage multiple people on the projects (as shown in cultural festival), she is very intelligent and can come up with solutions on the fly (like when they were searching for Rinne or when she had three-legged race with Kanon), she is a very hard worker (as evident by even Kanon's mom praising her later on, even offering her to aim for music university) and most of all she is very empathetic toward people. Her empathy being the one thing that carries her trough the story, being able to not see everyone else around as inherently evil and giving them a chance to explain themselves. Then helping them with their problems which in the end resulted in everyone being in a better place then when they have started (some examples: being stubborn in becoming friend with Kanon and listening to Ayano's story and offering her own opinion). She even influenced others to be like that (like when Kanon helped Tomita when she lost her hearing and when Ayano helped Kanon to bake and later on for two of them to come clear with their feelings).
Only times in the manga when she is showing deep psychological scars and any negative traits is when same-sex relations are brought into question. In middle school she panicked with Mahiru, when her piano teacher said she was pregnant she felt depression and loneliness, when Rinne confronted her about Kanon she fell to the ground in panic, when Kanon told her in joking manner that she might be in love with her she panicked. All of her fear and shortcoming come from the same issue - that she will lose everything once her own sexuality is in the light. That people will not only target her but that everyone who is close to her will abandon her. What we really need for manga to explore is how she came to that conclusion, as we have seen with people like Akira that she can be open about it and have a normal life. Akira might not have found herself someone that will love her but she can be open about her sexuality on her work with her colleagues and not having any negative consequences because of that.
Sorry for the long text, but TLDR is that I do not necessarily disagree with you that Saki might have other mental issues to work on. But when you look at manga as a whole there is not much that can you point out other than her internal homophobia. Moreover, Kanon has much bigger issues as she not only lost her hearing (which pretty much made her incapable to continue with perusing what she loved) but she was also ostracize by society (middle school when all the girls thought that she was only giving attention to boys) and she was betrayed by those closed to her (Ayano). It was Saki who was successful into bringing Kanon back from all of that trauma and enabled her to function normally again (at least in terms of having friends and goals in life again). Now it will be Kanon's turn to do the same for Saki.
What I can also agree is that I would love for manga to explore more of that concept, as we still do not know much about Saki's early years and how did she managed to get by. I really hope that this will be the highlight in the next few chapters as Kanon will definitely try to get more of Saki in order to understand her better. I just hope it will go in that direction and not end in 5 to 10 chapters now that they have kissed.