Anon you've lost your logic.
Firstly, it's really hard being a mother of three. Just in general. So put that to one side for now. It's hard. I only have one DC and when my DB was dying life was extremely stressful and I just found it so hard to stay calm with him when he was just being ordinarily annoying.
I can't see anything in your posts to suggest you are unkind. You worry incessantly about your DH and your DC. That suggests to me you are very kind. You show compassion and understanding towards your DM and DF.
You just don't feel lovable because you didn't get enough love as a child. The voices we hear from our parents and the behaviour they exhibit towards us are so powerful. We believe our parents emotionally even if not logically.
Your DH knows you better than anyone else in the world - he lives with you, has had sex with you, watched children grow in your body, has seen you every morning, woken up next to you, fallen asleep beside you, shared meals, seen your joy and your depths of sadness and numbness. And he loves you. He loves you and he knows you.
Go off and have a cry over first world problems. It's ok to cry. It's healthy to cry. If you can cry about what you're calling first world problems at least you can cry. Let those problems be the focus of your grief. It's healthy to cry, it really is. Crying is not a weakness it's an expression of emotion. Did you know that the physical shape and chemical composition of tears are different according to their origin? Emotional tears perform a function.
Fuck whoever told you you weren't allowed have feelings and express them. They were so, so, so very wrong. They couldn't cope with a child's tears and provide a bit of comfort - a reflection on them not you.
You are trying your best. You want out but it's because you want to be better. You remember that you loved being a Mum - because the depression wasn't taking over your life. This is a low point of an illness but it's not permanent. It is temporary.
It's hard getting through the days, it really is hard, and so tiring.
Have you been for therapy to talk about some of what has happened to you? Have you had any help to work through what your childhood and teenage years were like? I can still make my heart beat faster with anxiety right now this minute by focusing on a memory. I sometimes have nightmares and wake up sweating and have to remind myself I'm safe now and married to DH. And really nothing that happened to me was anywhere near as bad as you. I've been for counselling/therapy 4 times - as in 4 sets of counselling at different ages, not just 4 visits in total. My parents were arseholes. Doesn't mean I didn't love them or that I can't view them with compassion and forgive them, but I'm allowed be angry and call them arseholes (here and to myself) and acknowledge that they were selfish, wrapped up in themselves, and they failed me and caused damage. I can do that logically and say there's no excuse on one hand and be right and on the other hand acknowledge they loved me and couldn't see what they were doing, or couldn't face it, or weren't capable.
What are you on besides the diazepam? Personally I think diazepam is great stuff. You can fool DC with normal words and actions and not snap at them and even raise a smile even if it's a fake one.
Books I found helpful Adult Children of Alcoholics (have you ever been to Al-Anon?), The Anxiety and Phobia Workbook, Healing the Shame that Binds You, and Homecoming - Reclaiming and Championing Your Inner Child.
And what all of those books have in common is that they told me having depression and anxiety is not some kind of weakness or inherent fault, it's a result of shit things happening and not being able or allowed express hurt and pain. Having nobody to listen and care and support and getting through it on your own, learning not to trust anyone because nobody helps. But it's not any kind of inherent weakness, anybody else would be the same, and usually are although manifestations can vary. It's just a natural consequence of what the brain does. If you keep rubbing a stone it will wear away.
I remember going to my second counsellor and thinking oh my God, someone is actually going to help me. It was very helpful but I was still in denial about a lot of things and I still pushed a lot of stuff down. It was still engrained into me that things were not my DM's responsibility, and things were not my DF's fault as he'd had a rotten childhood himself. I couldn't look at my then relationship realistically because I was so used to putting up with shit and being the responsible one, until I broke from being too responsible and supportive to everyone else except myself. I was way too hard on myself.
The anti depressant I was put on was Lexapro which I think is called escitalopram, and it saved my life. 10mg first and then 20 quite quickly. For some stupid reason I thought taking the tablets was another sign of weakness but I couldn't cope any longer without doing something.
I no longer take responsibility for things I have no control over. I don't push myself to do things that will put me under pressure and cause me stress or anxiety. Normal people live like this all the time without even thinking about it. They just say no.
I am a kind person but my only responsibility is to myself and my DS. If I can do extra I will but not if it ties me or tires me or puts pressure on me.
And I consider going to the library a huge achievement to be honest. Getting DS to school, clean clothes, dinner every day, some kind of chat and play, my teeth brushed, his teeth brushed, playdates, my few hours of work, shopping done, baths and showers, clean house/not disgustingly dirty house, sheets changed, birthdays attended. They're achievements.
Sometimes we don't get to the library, sometimes we don't make it to hobby night. Sometimes we do. I do think I should be doing more but I'll do what I'm able. I need an enormous amount of time to myself in order not to be overwhelmed. So when DS was little I didn't do anything after he went to sleep at around 7/8. That was my time. Now that he's at school that's my time except for the days I'm working.
I realise I'm saying a lot and a lot is probably nonsense and unhelpful or irrelevant. I just feel for you because I remember how hard it is. 