There isn't anything unique about survival analysis that prevents individual prediction. Just like other regression techniques, you can make individual predictions. In fact, survival analysis often gives you something better: the full distribution of the duration! Let me explain.
Linear regression gives you an estimate for $E[Y_i|x_i]$, which is a summary statistic for the distribution of the random variable $Y_i | x_i$. If you did have the distribution of $Y_i | x_i$, you could compute the expected value, but also other quantities like the median, or some other business-influenced summary statistic. But alas, linear regression only gives you the expected value.
Survival regression, on the other hand, focuses on estimating the survival function (what you call survival probability over time). The predicted survival function is an estimate for $P(Y_i > t | x_i)$, which has the same information as the distribution of $Y_i | x_i$. Hence we can choose the summary statistic, like $E[Y_i | x_i]$, or the median, or some percentile, etc.
expected time-to-event for individual
so it sounds like you want the expected value. Either you have to compute this from the survival function, but often the software does this for you. In Python, two packages that can do this are lifelines and scikit-survival.