I think your last sentence is correct, @MARattigan! 
For 1. a4 you need to deal with one response for black that you can prove forces a draw. That's all. (Assuming 1. a4 is not in white's strategy and chess is a draw).
White's strategy is one move for each possible position that he could see. Black's strategy is similar.
So White's strategy deals with 1 initial position and 39 positions after black's first move (see reasoning in previous post). This makes no assumptions about black's play.
Likewise, Black's strategy deals with all 20 positions after white's first move but far fewer than the total number after his second (because he only needs deal with positions reached by strategy moves by him). And so on.
The fact that both players reduce the positions like this means that a position that can only be reached by both players playing at least one non-strategy move can be ignored, since neither strategy requires it. For example if 1. d4 d5 are strategy moves, the position 1. e4 e5 can be ignored.
A weak solution of this position, for example would have to look something like the accompanying moves.
The game theoretical value is a draw, so the Syzygy tablebase would not do for White's strategy according to van den Herik's definition, because it would produce a win if Black played anything but KxN and the definition insists on a draw.
I'm not saying that the moves I posted don't constitute a weak solution. I would say, however, that the moves Syzygy gives would also constitute a weak solution, to the man in the street and also to people working in the game theory arena.
It may depend on definitions used implicitly. The game theoretic value of that game is a draw. A weak solution provides the game-theoretic value of the initial position and a strategy that guarantees at least that value (actually "at least" is even redundant, because the game-theoretic value is the minimum outcome that can be achieved with optimal play). So anything White plays after any first Black's move is indeed an optimal White's strategy for a weak solution, because Black cannot win. But since Syzygy tablebases are a strong solution, the game-theoretic value of all the legal positions from start to finish and a strategy to achieve that value are known, so If Black does not capture, the position becomes a win for White and an optimal White's strategy for a strong solution must achieve victory. Nontheless, since White cannot lose, her strategy for a strong solution is also good for a weak solution. So yes, there are implicit nuances in Allis and van den Herik's definitions that for practical purposes it maybe better to explicit.
I add that if an optimal strategy for White is known, indeed it is also known what is the best opposition Black can offer. In fact, the optimal strategy for White must work for all Black moves, so we know that the best opposition she can offer is the capture, in that position; therefore, we can determine an ideal game, where both sides do their best to achieve the draw (the game value of the starting position).