But we DO find the screen, as a whole, very useful and an excellent addition. So, what we are looking for is quite different.
We just want to change parts of it on a multisite/subfolders network.
To be exact. The wp logo image (wp-badge.png) on the left, as well as the header next to it (.welcome-panel-content .about-description, .welcome-panel h3) and the text under it included in (.about-text, .about-description, .about-wrap li.wp-person a.web).
I am aware that most of it is hardcoded. Still.... There must be a way. :)
PS: 1) We already use a plugin that can add css on the admin screens. But I don't think it would solve anything. 2) If there is a way, please be as specific/descriptive as possible. My coding skills suck. :(
After following brasofilo's instructions I have compiled the code to a mu-plugin. I placed it in our mu-plugins folder, have tested it on our test-multisite and works like a charm. :)
I have it uploaded to my dropbox so anyone can benefit. Feel free to Right Click Here, download it and customize it to your needs. I hope it helps others as well.
PS: @brasofilo Please let me know whether I have credited it correctly.