@Eljuliovoltio Hello,
Thank you for your detailed post, from the photo range extender does not obtain Internet from router or failed to connect to it.
Please confirm RE305 is on the latest firmware (select hardware version first to download corresponding firmware)
https://www.tp-link.com/support/download/re305/
May I know how about the LED status on the range extender after the set-up?
Here is a guide to troubleshoot RE installation step by step, kindly give it a try and let us know if it works for you,
Has connected to TP-Link extender but no internet, what can I do?
https://community.tp-link.com/en/home/forum/topic/198516
Please click on the Status with exclamation mark and check if 2.4Ghz/ 5Ghz failed to connect, and double check Wi-Fi password there.

If still the same, may I know how about the LED status on range extender after installation, what is the model of your Router, can I have a photo of the wireless settings on main router? You could try to login web page of the range extender at that time, go to Settings -> Log, save and share the log file to us :
https://www.tp-link.com/support/faq/1398/