I routinely run with only direct cookies permitted (no third party ones) and with all cookies except those I have explicitly whitelisted being deleted each time my browser is closed.
I am not aware of any problem this has caused me for a long time, including on the sites you mentioned. Maybe there is some useful feature I'm not seeing at all because of the cookie restrictions I impose, but maybe they've just got better over time at not relying on cookies for things they shouldn't?