Hi there the_gambit,
It sounds as though you're receiving a message regarding needing to allow cookies even though cookies is enabled within your settings. You're in the right place to seek a solution and I'd be happy to assist.
I understand that you've made sure to check that the "Block All Cookies" option is turned off and that you have attempted to clear history and website data as well.
Let's check to see if "Prevent Cross-Site Tracking is enabled as it can sometimes cause behavior such as described.
To check, tap Settings>Safari>Prevent Cross-Site Tracking.
If the issue still persists after turning off the "Prevent Cross-Site Tracking" feature, Profiles can sometimes cause behavior such as described. Below is a resource that provides more information on Profiles:
Install or remove configuration profiles on iPhone
I hope this helps,
Cheers!