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I am going to get a finite sample (size $2n+1$) of data from some process which is not symmetric about the median. I want to use either the mean or the median as a summary statistic.

If I choose the mean, there is some distribution of the sample mean based on $n$ samples. I can take solace in the fact that the average value of this distribution is the same as the true mean (sample mean is unbiased). However, if I take the median of the sample means then that is not equal to either the true mean or the true median.

On the other hand, I can choose the sample median and there is some distribution of it based on $n$ samples as well. This time, I can observe that although the sample medians follow some distribution, the median of that distribution is the same as the true median. On the other hand, the mean of the medians isn't the true median (the median is unbiased).

My goal is to estimate something "well". Meaning I don't want systematic difference (admittedly a vague criterion). Should the unbiasedness that the sample mean offers or the fact that the sample median is the same as the true median be the preferable?

I understand the answer can be "it depends". In that case, I'd appreciate concrete scenarios where one or the other would be preferable.

One scenario: I pay a penalty every time my estimate is different from its true value with the magnitude of the penalty being proportional to the amount by which I'm off. My intuition says that unbiasedness should be something to strive for by this criterion.

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First, you should read (perhaps beginning here) about the "Central Limit Theorem for Medians."

This theorem is often proved and illustrated along with 'order statistics' in mathematical statistics texts because there are also CLT's for other order statistics (except for the max and the min).

Provided that the population distribution density is positive at the population median $\eta,$ sample medians $H$ from sufficiently large samples are approximately distributed as normal with median $\eta.$

In particular, consider a severely right skewed population distribution such as exponential with mean $\mu=1$ and median $\eta = \ln 2 = 0.69315.$ Medians $H$ of samples of size $n=1000$ will be approximately normal with mean $\eta.$

This is illustrated (approximately) by the following simulation and figure:

set.seed(2021)
h = replicate(10^3, median(rexp(1000)))
summary(h)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.5981  0.6721  0.6940  0.6934  0.7137  0.7931 

hist(h, prob=T, col="skyblue2")
 curve(dnorm(x, mean(h), sd(h)), add=T)

enter image description here

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