Intuition: Why chance of getting at least one ace when rolling a dice six times is not close to $1$?

Think of it this way: roll the die a large number of times, and break the rolls up into groups of $6$. We expect that roughly $1/6$ of the die rolls will be aces, so on average the groups will contain a single ace. But some groups will have two aces. In order to have an average of 1 ace, for every group with two aces, there has to be another group with no aces. And of course for every group with three aces, we need two more groups with no ace. Etc.

When you realize how common multi-ace groups will be, it becomes obvious that there also has to be a lot of groups with no aces to balance them.

When you roll a die six times you shall expect to obtain one ace.

This is an average result, so the probability for obtaining zero aces should counterballance the probability for obtaining more than one aces.

Thus your intuition should anticipate that the probability for obtaining at least one ace is not too much more than one half.

The intuitive explanation is that there are $6^6=46656$ different outcomes from rolling a dice 6 times, and $5^6=15625$ of these have no aces. $15625/46656 \approx 33.5\%$, so $33.5\%$ of outcomes have no aces and $66.5\%$ of outcomes have at least one ace. If the outcomes are equally likely then the probability of at least one aces is the same as the ratio of the number of favourable outcomes to the total number of outcomes i.e. $66.5\%$.