No https on old Android

I have an old Android tablet (4.0.4) that no longer can be updated and no browser can display https sites, which is almost any nowadays. I have tried many browsers, they all fail. There is no error message, all browsers complain that I have no Internet connection. I do. Other apps and plain http sites work fine.

Is there anything I can do about it? Is there any browser that will do https independently, without any support from the OS?

I am ruling out alternative firmwares such as CyanogenMod or anything like that. I have bricked a phone with one of those, asked for help on the forum.xda-developers forum and was duly ignored.