Intro:

This solution can now be used in SharePoint On-Premise Environments only. If you are working in SharePoint 2013 or SharePoint 2016, this article is for you.
For SharePoint Online and SharePoint 2019, there is a new feature that fixes this problem. It is available starting from ShortPoint SPFx version 6.5.10.21 and will be available in all newer versions.
Our On-Premise environments users will soon also have access to the feature mentioned above.

Issue:

I would like to connect to a SharePoint List / Library on another Site / Site Collection.

This Site / Site Collection is not shown in the pre-populated drop-down of available connections:

Reason:

The list of sites and site collections appearing in the drop-down depends on the SharePoint search API. There are certain cases when a particular site is not displayed in the list.

1. If you want to connect to a Modern site, please make sure that at least 24 hours have passed since the Modern Site was created. This may also happen with Classic Sites and new Site Collections.This time is required for SharePoint Search Service to crawl the data about the newly created Site / Site Collection and make it available for you in the drop-down.

2. In all other cases, please proceed with this solution.

Solution:

Prerequisite: make sure the latest version of ShortPoint is installed in your environment. This solution will not work on old versions of ShortPoint. Tutorials on how to upgrade to the latest version here: for SharePoint Online and for SharePoint On-Premise 2013/2016.

Step 1

Open the Site / Site Collection that is missing from the drop-down of available connections.

This is the Site / Site Collection in which you have a SharePoint List / Library you would like to connect to. In my example, I'm using a regular SharePoint List called Target SharePoint List. Here is how it looks like:

Step 2

In that Site / Site Collection, create a new wiki page.

Step 3

Into this page, insert a ShortPoint element that you are trying to connect. I will be inserting an Image List:

IMPORTANT: The Site / Site Collection in which you are now working might not have ShortPoint installed/activated yet. If in your newly created page you do not see the ShortPoint Insert buttons, please:

for Office 365 (SharePoint Online) users: add ShortPoint to the site contents and run installation of ShortPoint from Dashboard.
for SharePoint 2013 / 2016 (On-Premise) users: proceed with activation of the ShortPoint Site Collection Feature / Site Feature as explained in part 2 / part 3 of this tutorial.

Step 4

Open the setting of the ShortPoint element you have inserted by pressing the cog wheel as shown below.

Switch to the Connect tab and do the following:

In Select Connection Type, choose Other Sites. Do not choose Current Site.
In the Site Collection, choose Current Site Collection.
In the Site, choose the name of the site you are currently on (in my case it is called Second Site Collection but this is the site name)
In the List / Library, choose the name of the desired list.
In the View, choose the View you would like to use.

Step 5

Map the items as explained in this example (step 4). Update the element after you finished with mapping. You will see it on the page, and it will now have a relative link, looking similar to the one below:

If your element link does not look like this, please go back to the Prerequisite of this article.

Step 6

Copy the element as shown below. Return to the site collection you are building your page on. Paste the element.

Because now it has a relative link in it, the element is connected to the Site / Site Collection you could not find in the drop-down, which was our original objective: