We're gonna see what we can figure out about how this game works, or at least speculate wildly to that end. Fair warning, long, dubiously informative, and largely pointless post ahead
If we assume that this chess variant with similar rules to chess save for piece movement (fairy chess) and that the king has fairly constrained movement, then the fact that white sets up first means the game will almost always (and definitely in this example) be a black win- as it is far too easy to place pieces to checkmate the king from before the game starts.
Now, our first of many assumptions is that this game is not boring. That is to say, if a ruleset leads to a game with shallow strategy, then it's probably not the right ruleset. Notably, if a ruleset would lead to obvious forced wins from the start, then it's probably not the right ruleset.
Our second assumption is that this game is like chess. Obviously it's different in multiple ways, but a ruleset which is similar to chess is more likely to be accurate than a completely divergent ruleset.
Our third assumption is that we haven't cheated. That first placement was apparently valid, meaning we can place pieces on top of other pieces.
Our fourth assumption is that the rules are elegant. Simplicity is preferable, and rules should not frequently have exceptions. Notably, few pieces should have special behavior outside movement patterns. This is the most dubious assumption, and the most likely to discard the true ruleset.
The most obvious ruleset would be fairy chess on a triangle board, except this would violate assumption 1, and besides we've seen that two pieces can coexist on the same space. Still, it's a good baseline.
To allow for our king to be on the pawn, we have to assume that the pawn does not automatically capture the king nor vice versa. Perhaps we need a certain level of unit to capture another unit- but if that were the case our unit's levels should have been listed. Perhaps this is special to the king, and he cannot capture pieces but can move to share their space. Or, perhaps this is merely a quirk of the set up phase and won't be repeatable once the game starts.
To fix the "win from setup," we either need to make the king able to escape easily, or have an alternate victory condition.
If we choose the first option, there must be a way to win after setup- and for black, setup seems to be the easiest chance for a checkmate. To fix this, there must be some way to gain material throughout the game, making checkmate a matter of gaining enough material to threaten most of the board.
The second option... has so much breadth that it's hard to say much about it. Andrew's setup would indicate that the white king, or possibly square (1,1), needs to be defended- but he could also just suck.
Right now, my two favorite contenders for ruleset are:
King can move to any square, but cannot capture. Convert enemy pieces on capture, and up to the point where you can put the entire board under threat. This does raise some questions about threatening squares which have one of your own pieces in them, but that's easy enough to work around. Unfortunately, black's massive advantage from setup is still present to some degree.
Victory condition is asymmetric- for black, you must get the king to (1,1). For white, you must get the king to (1,11) or (6,1). White's first setup disadvantage is mitigated by having two possible destinations. King cannot be captured, but cannot move into threatened squares. This is to allow for defence, without allowing black to instantly checkmate white king. This has the advantage of making Andrew's setup somewhat reasonable, though why he'd put his king so far back is a mystery.
Of course, there's always the option that this is fairy chess and Andrew just sucks enough to let us get a turn 0 checkmate.