Experience shows that a new kernel release can be pestered with issues and unless you are a first-adopter - with all the perks that goes with that - you should never hurry onto a new kernel.
You can see the state of the kernels on https://kernel.org as Manjaro follows this schedule on unstable branch.
As it is unstable branch which get the new kernel - there is no way of telling the user of a Manjaro system that a new kernel is available or EOL - but you can only get that version by switching to unstable branch.
When a kernel is removed from Manjaro repo because it is EOL - then it cannot be installed anymore - but the user is responsible for updating the system.
The popup is not a command - it is an information.
Have you checked the settings for the settings-notifier? Rightclick the icon in the taskbar and open the settings. Or open Manjaro Notifier Settings from the app menu.
The reason you are getting the message is because your Notifier Settings is instructed to inform you when a new kernel is available but it is not mandatory to update.
You only mention 5.10 and 5.13 but there is a newer kernel version 5.14 available.