k8s-backed cluster remains active when client's kernel dies unintentionally

**What happened**:
I'm not sure if this is a bug report or feature request, but the behavior is not desired for our use case and I can't seem to figure out a way to get the desired behavior. The issue occurs when the kernel for a notebook containing a client (which is linked to a `GatewayCluster`) unintentionally dies (e.g. due to a memory overload issue). We're using the kubernetes backend within a `daskhub` helm chart configuration. Even if the cluster has been created with `shutdown_on_close=True`, the scheduler does not seem to shut down in these instances and instead remains active until its `idle_timeout` setting is reached and can be reconnected to from a new notebook. This does not occur if the cluster is shutdown explicitly, nor does it occur if you intentionally shut down (or restart) the kernel.

**What you expected to happen**:
The behavior I expect (and desire) is based off of my experience with `dask-kubernetes`, in which I was mainly using a local scheduler. In this case, the scheduler was virtually always in the same kernel as the notebook, so when one died, the other died. I recognize the `dask-gateway` remote scheduler situation might be a bit more difficult to handle, and that in some cases you would want your scheduler to persist in a situation like this. But I'm wondering if there is a flag that could be used to signify if you want to shut down a cluster even if a client connection is closed _unintenionally_.

**Anything else we need to know?**:
If this is just a matter of implementation, I'm happy to take a crack at this if others have a sense of where to start. I did a bit of digging but didn't have a great idea of where this behavior might be altered.

**Environment**:

- Dask version: 2021.4.1
- Python version: 3.8.8
- Operating System: Linux (`daskhub` helm-chart defining a kuberentes setup)

No file or test is named in the issue. Start by reproducing the Kubernetes-backed GatewayCluster case with shutdown_on_close=True and an unintentional client kernel death, then compare it with explicit shutdown and intentional kernel restart. Done means the expected cluster lifecycle is defined and verified for the reported scenario without breaking the documented persistence behavior.

Written by the indexing model from the issue text.