Was the danger of radiation in the case of nuclear war overestimated during the cold war?

Yes, the effect of radiation was significantly over-estimated (it still is by most people). But it was still bad enough to ruin quite a few days.

The worst immediate effects of a nuclear bomb are the blast and heat damage. Immediate radiation is also quite dangerous for people close to ground zero who survive the blast and heat. Many such survivors will die of radiation poisoning in the next week or so.

As far as I know, none of this was significantly mis-estimated. (Hint: it was really bad for people within a mile or so of ground zero.) What did turn out to be overestimated was long-term effects of radiation such as cancers in later life for people who were nearby and survived the blast and its immediate effects or who were more distant.

As far the as the huge doors go, they would appear to be primarily aimed at preventing blast damage. Assuming they were well-engineered, they'd probably work pretty well.

Once the blast and its immediate effects are over, residual radiation declines rapidly -- it has to, because high radiation means a rapid rate of radioactive decays which means a short half-life.

But to be clear, what's left is still very dangerous! (If I were close to ground zero in an attack, my own preference would probably be to stay in the shelter for up to a week and then get out and try to get out of town.)

All in all, what you describe sounds like the sort of panicky over-reaction we saw in the US in the 50s and 60s, but with a bigger budget behind it.

The main question I see is the issue concerning 'Keeping people underground for days'

The issue which these civil defense type shelters were looking at isn't just the immediate local contamination due the nearby blast. It is what we call fallout. Every particle in the region of an explosion is sucked into the blast area, and then projected upwards within the classic 'mushroom cloud' we see in so many pictures. These particles are radioactive, and they take time to come back to ground. A single line of the linked Wikipedia article has an estimate on this effect (emphasis mine):

Fallout radiation decays relatively quickly with time. Most areas become fairly safe for travel and decontamination after three to five weeks.

This is the reason the civil defense plans were accounting for an extended stay underground. We can track the source from the linked wiki to an article from 1957 on The Nature of Radioactive Fallout and its Effects on Man relating to estimates concerning radiation belts formed worldwide after the nuclear testing being done in the 50's:

The second type stropospheric fallout consists of that material injected into the atmosphere below the tropopause which is not coarse enough to fall out locally This debris is sufficiently fine that it travels great distances circling the earth in the general latitude of the explosion until removed from the atmosphere by rain fog contact with vegetation and other meteorological and or physical factors The average tropospheric fallout time is estimated as 20 to 30 days The fraction of the fallout which is in this category depends mainly on the size of the explosion and the conditions of firing If the explosion exceeds a certain minimum size about one megaton MT the fireball will have enough energy to penetrate the tropopause carrying fission products into the stratosphere Smaller detonations leave in the troposphere all debris not deposited locally The fraction of the fission products from a large weapon that remains in the tropopause depends on the size of the explosion conditions o firing and meteorological factors

Remember this circulation of radiation which would last 20-30 days is cumulative. Every bomb detonated would contribute to it. And it would continually fall out of the atmosphere to the ground below. This is the radiation hazard the designers of your civil defense system were building to accommodate. All the atmospheric radiation for every bomb detonated worldwide would be circulating for up to a month after the nuclear war they were trying to plan for.

This is the history of what the system was designed to handle. The question of did they overestimate, thankfully we don't know. Any guess there would be speculation.