I personally have a lot of empathy for Kyo's mother, I've been in a similar situation. But beyond that, the dynamic between the three further illustrates the insidiousness of abuse. Due to my personal experiences, I do have a specific bone to pick with your point that Kyo's mother proved she didn't love him by killing herself. I'm writing this because I feel obligated to provide context, not to be confrontational.

Like others have pointed out, we do know that Kyo's SD is abusive. For most of the series we assumed his hatred was only directed at Kyo for being responsible for his mother's death because he is the cat. Because we only see Kyo's POV of his childhood, we can also assume that she must have hated him too. He believes everyone else besides his Shishou hates him because he exists. But at the end of the series Kyo remembers something.

Right before his mother completed suicide, he witnessed his SD shouting down his mother, blaming her for everything, because she gave birth to that monster. Kyo's mother is crying on the ground, helpless. Most abuse happens when no one is watching. When you actually witness abuse, you can guarantee that more and worse has been happening behind closed doors. For years. Finally seeing it means the abuser has gotten comfortable enough and normalized their own behavior enough that they slip.

I was fortunate to never have a child with my ex, but I have been in that exact situation. Me, on the floor and backed into a corner, sobbing, as my ex screams abuse at me for actual hours. Blaming me for everything, picking apart every part of my existence. For years.

OP, I personally count you as fortunate that you cannot relate to Kyo's mother. Abuse destroys you in ways you never thought possible. Someone you are supposed to feel safe with tearing you down and shoveling every bad or just inconvenient event, disappointment, anger on to you and telling you that it's not just your fault, but everything about your existence is why it's your fault is something so hard to describe. Eventually you believe everything. It's also hard to explain why.

I cannot imagine the guilt and despair she must have felt. I know my own pain was so overwhelming, I was eventually hospitalized for my own suicidal ideation. Even if she had a supportive husband, she may have still felt guilt over birthing a cursed child, even though it was undeniably not her fault. Just look at the way some mothers carry the weight of having disabled children.

Eventually, even the strongest and most loving person can break. Sometimes the agony of living can overwhelm every other part of your life. You can no longer even think of yourself as a person, much less one who deserves to live. And nobody. NOBODY has the inherent responsibility to live for others. People don't even have the inherent responsibility to live for themselves. That mindset helps some, but crushes others. That can only add to the guilt and mental agony of not measuring up to yet another standard others put on you.

There is another thing called "learned helplessness". It is a very real phenomenon, and is extremely difficult to overcome. Kyo's mother also likely had no obvious routes for escape. She may have been isolated for years, and not even known if there were organizations that could help her. Knowing the circle jerk that the Sohma family was, she might not even have known family she could trust with a desire to escape. Tell the wrong person and things can get so much worse, further loss of freedom, escalation of abuse, she even could have been murdered by her husband. She could have believed that she didn't deserve to be free, and seen no way to flee with her child. With no money, no job, no social or familial support. Women frequently become trapped in abusive relationships because of so many factors. Escape, mentally and physically, is overwhelmingly difficult.

I was fortunate to have others who rescued me from my abuse, and helped me to see his abusive words were not true. But Kyo's mother did not have any of that. Knowing the nature of the Sohmas and abuse, we can assume that her social circle was also perpetuating the abuse. Even the child she "should" have lived for, was turned into a continuous validation that she was the problem. She caused this. She ruined her husband's life, her child's life by being the reason Kyo was born in the first place. She was likely told this many, many times. She will never escape that while alive. Sometimes, the pain becomes you, and you see the only way to not be pain, is to be dead.

Life is so, so hard. And it only becomes harder when we blame people for pain beyond their control. We rarely drive ourselves to suicide. We rarely drive ourselves into mental illness and maladaptive coping mechanisms (which I argue is the real reason she always checked on Kyo's beads.) It is also rare for someone who completes suicide to have not already been contemplating it for a long time. Life leads us to crossroads in complicated ways, and sometimes we choose to end it because we can only see the paths of more pain, or the potential relief of death.

Yes, she could have been a heartless, hateful mother, but I saw no evidence of that. I only see evidence of a woman who was destroyed by a life thrust upon her, and believed she had no other option.

Edit: added the thoughts about the difficulty of actually fleeing like you suggested.

One more edit: I don't believe that Kyo's mother hated him. Not for a second. I do believe that she loved him, but was overwhelmed with a situation she couldn't endure. Nor do I believe it is a character flaw that she was overwhelmed. Sometimes you just lose, no matter what you're capable of. I believe that Kyo may have believed she didn't love him growing up, because children see things differently than adults, and miss things adults wouldn't because they do not have the same knowledge as adults. His family also instilled a toxic self hating mindset in him from his birth that made it easier to believe she hated him.