Are neural networks prone to catastrophic forgetting? Imagine you show a neural network a picture of a lion 100 times and label with "dangerous", so it learns that lions are dangerous.

Now imagine that previously you have shown it millions of images of lions and alternatively labeled it as "dangerous" and "not dangerous", such that the probability of a lion being dangerous is 50%.

But those last 100 times has pushed the neural network into being very positive about regarding the lion as "dangerous", thus ignoring the last million lessons.

Therefore, it seems  there is a flaw in neural networks, in that they can change their mind too quickly based on recent evidence. Especially if that previous evidence was in the middle.

Is there a neural network model that keeps track of how much evidence it has seen? (Or would this be equivalent to letting the learning rate decrease by $1/T$ where $T$ is the number of trials?)
Yes, indeed, neural networks are very prone to catastrophic forgetting (or interference). Currently, this problem is often ignored because neural networks are mainly trained offline (sometimes called batch training), where this problem does not often arise, and not online or incrementally, which is fundamental to the development of artificial general intelligence.
There are some people that work on continual lifelong learning in neural networks, which attempts to adapt neural networks to continual lifelong learning, which is the ability of a model to learn from a stream of data continually, so that they do not completely forget previously acquired knowledge while learning new information. See, for example, the paper Continual lifelong learning with neural networks: A review (2019), by German I. Parisi et al., which summarises the problems and existing solutions related to catastrophic forgetting of neural networks.