Linear message-passing models have emerged as compelling alternatives to non-linear graph neural networks for unsupervised node embedding learning, due to their scalability and competitive performance on downstream tasks. However, we identify a fundamental flaw in recently proposed linear models that combine embedding aggregation with concatenation during each message-passing iteration: rank deficiency. A rank-deficient embedding matrix contains column vectors which take arbitrary values, leading to ill-conditioning that degrades downstream task accuracy, particularly in unsupervised tasks such as graph alignment. We deduce that repeated embedding aggregation and concatenation introduces linearly dependent features, causing rank deficiency. To address this, we propose ACC (Aggregate, Compress, Concatenate), a novel model that avoids redundant feature computation by applying aggregation to the messages from the previous iteration, rather than the embeddings. Consequently, ACC generates full-rank embeddings, significantly improving graph alignment accuracy from 10% to 60% compared to rank-deficient embeddings, while also being faster to compute. Additionally, ACC employs directed message-passing and achieves node classification accuracies comparable to state-of-the-art self-supervised graph neural networks on directed graph benchmarks, while also being over 70 times faster on graphs with over 1 million edges.