Anna’s proposal should work, but here is the more fundamental reality.
The list of available datasource actions is built based on the model context of the component that is firing the action. So if you button’s container is not bound to a model - we don’t know for sure what actions should be made available. And because the available SFDC WebLInks are sourced from metadata about the sObject - we need that model context in order to retrieve the availalbe webLinks. So we don’t show the SFDC WebLinks option…
So - make sure your button set (or table or page header) is tied to the model where the SFDC Weblink action is configured. And the action should be available.
I admit - we could do better here. I think we should allow weblink actions to be selectable even from a button that does not have model context - as long as you enforce setting the model in the configuration of the button. We’ll add that to the backlog.