Imagine this polarized set of votes. 0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9
The median is 2.
Someone who has a lot of votes, he keeps two of his votes and cast them the last minute.
0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9
Now the median is 8.
In a polarized set of votes, the one who has a lot of votes can manipulate the result whithin the bounds of the polarization.
This cannot be done in the mean average.
I appreciate you coming around to realise that vote manipulability is a real issue and we need to avoid it. The problem is that with the average you just need a single person voting 125. But of course people with multiple votes can make their vote stick out less. But here are a few points about your counterexample
you always have counterexamples
Yes, absolutely. One of the thing that you notice when you study Voting Theory is that if you take pretty much any two voting system you can always design a problem ad-hoc that will come up with different result for the different voting systems. And always some special cases where it makes more sense one voting system respect to the other. There are books written just with the list of those counterexamples, and the geometry of the various voting system.
Here is another counterexample I placed in my book if you are interested (conflict of interest warning, the link goes to a Patreon page from where you can support me... not on Dash unfortunately).
if the group is naturally polarised the median gives the power to the bigger group
So, yes, if you have a group that is naturally very polarised and that do not spread through the whole length of the space, then as soon as one of those poles reaches more than 50% of the votes they immediately have the winner. The other group can do anything they want, but they are, de facto, a minority. And thus counts nothing. We said the winner is the Condorcet Winner, and if more than 50% of the people vote for one candidate over all others, that is the Condorcet winner by definition. No other questions needed or asked.
And if the two poles are exactly of the same strength, and if the median voter realises that they are the "weighting needle" (translated from the Italian, ago della bilancia, meaning the person in the center that chooses where a situation goes) now that person has a huge power. But this is ok because it is a very rare situation, And it is very difficult for a person to design such situation. Most of the time the situation is not naturally polarised and if it is polarised the two poles do not have exactly to the last vote the same weight.
LATE EDIT: The study of the relative power of the various groups in decisions is something very much studied. I also have a chapter in my book about it (the one on Banzhaf Power Index), but it is not ready for public consumption (the chapter, not the studies

).
The problem with people having multiple votes
Then there is the problem of people having extra weight. And I think here you struck (finally I must say

), on a real limit which is in Dash. All the experiments that are done with "The wisdom of the crowds" are done with a situation where each person has only one vote. And there is also a paper showing how as soon as people can influence each other, the result goes astray. Having multiple votes is the ultimate form of influence. One person literally controls more than one vote.
So, yes, this is not a good idea if you are trying to extract the wisdom of the crowd (an approximation of what an expert in that would say, without knowing who to ask to, and when many "experts" tell you different things). And absolutely yes, if you have more votes to play with you can manipulate the median. Basically the median looses its great properties that it had respect to the average. If someone controls two votes, now the result is within an error of 2 people from the median voter. If someone controls n votes, you know the right answer is within n from the median voter. Make n big enough and the vote is irrelevant. If n > 50% you can avoid voting, and just ask this person, and the result will be irrelevant. The only thing you can say on the other side is that a person that has more votes has a bigger incentive to see this project succeed, so they will make more research. But there are wonderful examples of situations in history where this was not enough
The Comparison with Prediction Markets
The only thing I can think of, is that in Prediction Markets you have the possibility to have "voters" (people that bet, "bettors" ?) that have different weight. And the person receiving the bets must adjust the weights so that both sides have equal value. Only in this way they do not lose money. And where they stand at the end is the best prediction of the outcome.This is very much like a system where people use the median. It is also like a system where people use the average... if they care more about their personal turnout and not about the collective result. It has been shown that prediction market are on average a good way of predicting the outcome (reference available)
Differences with a Prediction Market
But here we have a system where people care about the collective result (they don't gain something by being right against everybody else), and where some people have more weight than others. hmmm. Not necessarily a recipe for success. And the reason why it took me so long to invest in Dash, and the reason why if this shortcoming does not get addressed I might look for alternatives. I do have a suggestion that I will present in the next days. I don't expect the suggestion to pass, but it is something that will given an economic incentive to people to keep all their dash together. Thus exchanging the bigger weight in the online senate with some extra Dash.
So, to summarise:
The median does not have strategic voting.
the average has strategic voting.
Iterative Strategic voting polarises the votes if you use the average but not the median.
If the system is naturally polarised, the median just gives the result to the biggest block (if two poles). In that case, sometimes it is better the average but this case is rare and then you are better off using a different form of voting.
People having multiple votes can make the result less precise.
Partially the fact of having a bigger stake makes this less a problem, but it is still a problem.
This is not a Prediction Market.