Leela is more or less entirely NN-driven, if you ignore the vast influence of the human-directed experiments and decisions that resulted in the underlying architecture we have today. But yes, Leela uses its neural network both as an evaluation function and to generate a "policy" that guides the MCTS-like process by which the game tree is traversed.
There's no fundamental reason why Stockfish's NNUE couldn't be modified to additionally output a move ordering, decisions on pruning, extensions, and reductions, etc. But you would probably want to make use of at least some of the extra input features and processes currently used in the handcrafted search heuristics, like SEE, MVV-LVA, various killer/countermove/history tables, various statistics like depth and aspiration windows, quiescence, just to name a few. So it definitely wouldn't be anywhere near as simple as integrating the NNUE evaluation function into Stockfish was.