But how could we make a link that doesn't choose the search engine for the user, and lets the user choose which one they prefer? Is there a service that does this? or even better something that doesn't require any third-party, maybe some javascript library?
edit: The same could be applied to maps, like Wikipedia's external map chooser: a link that would let the user choose which map service they would like to use instead of sending them to one by default.
There used to be a external.IsSearchProviderInstalled native JavaScript method[1], but it was deprecated by browsers, most likely to prevent them detecting and recommending one search provider over the other.
Simplest workaround will be to show a dropdown select list or page modal with search engine options, for the user to be able to select their choice.