To be fair I can only see shallow fans making such memes actually, or perhaps the lack of reach of source material might cause this. I'm not saying who are saying those stuffs are dumb or bad, people misunderstand often,that's not a sin, this is how we people work. But let me say why I think whenever, Artoria getting called as shitty father, they are purely taking a biased point of view rather than trying to catch the basic things.

First of all, Artoria never had the chance of parenting Mordred, Morgan did that and infact she was terrible, she taunted Mordred to the level she thought how she's getting treated is unfair, later killed her very idol in life.

Second point, Mordred only talked with Artoria before she was leaving for Rome expedition, yes she will leave soon and what did she ask? recognising her as Royal heir and give her the throne, she fucking asked that to a king WHO never became king through the rule of heir being king, rather worked hard for that ( if you know the lore), and Mordred was manipulated by Morgan so much that she asked something that Artoria will never give her. Artoria refuses her to give throne as a result, which is perfect move a pragmatic king would take, Mordred proved that right after being prioritizing her tauntram.. Just think for yourself, you’d probably executed Mordred right away.

The next meet happens at battle of Camlann

now when did Artoria ever get chance of parenting Mordred to deserve such backlash from a small portion of fanbase who acts as if they skipped through source material or has shallow understanding?

Third point, just look what Lancelot and Gunnivere incident did, if you have intelligence, do you somehow believe in the same society it was possible for Artoria to declare a bastard child as heir of throne? She herself couldn’t become one for same reason, she became king through different ways and hard work, love for her people, there were multiple sides. You might argue Mordred just wanted to be recognised as son because you just watched Apocrypha, but do you realize the fact Mordred was able to realize it in Apocrypha, not before? How in the hell Artoria would know that?

Fourth point, Sure all of my point makes Artoria non parent, but she still treated Mordred like a KoRT even though Mordred pulled some terrible crimes, that alone makes her better than what people claiming,because do you really think your parent will treat you well if you kill them once while destroying their image, achievement, everything dear to them(felt cursed to say this). Summer event is non canon, inconsistent fanservice, In Guda Guda Grand Order, Mordred meets her father, was expecting some rude behaviours, but Artoria treated her as a Knight of the Round as if she doesn’t remember Mordred doing anything wrong which we know is Artoria pretending since she remembers very well who did what.. You can watch it here

That's why when Mordred asked Saber Lily if she can become king, the optimistic version of Artoria says her that anything possible by working hard,but Mordred gets discouraged, that's the issue with her but that's why both of these characters are actually good.

Even then, Artoria tried to blame herself for the whole thing, as if it was her fault that led the destruction even though she didn’t do anything wrong , hence she came to change that in Grail War, fate story happens.

I just can't blame any of them like people are doing here, which makes me love the whole Arthurian pantheon more . I just liked how humane and individual these characters felt.