The emergence of SARS-CoV-2 variants stressed the demand for tools allowing to interpret the effect of single amino acid variants (SAVs) on protein function. While Deep Mutational Scanning (DMS) sets continue to expand our understanding of the mutational landscape of single proteins, the results continue to challenge analyses. Protein Language Models (LMs) use the latest deep learning (DL) algorithms to leverage growing databases of protein sequences. These methods learn to predict missing or marked amino acids from the context of entire sequence regions. Here, we explored how to benefit from learned protein LM representations (embeddings) to predict SAV effects. Although we have failed so far to predict SAV effects directly from embeddings, this input alone predicted residue conservation almost as accurately from single sequences as using multiple sequence alignments (MSAs) with a two-state per-residue accuracy (conserved/not) of Q2=80% (embeddings) vs. 81% (ConSeq). Considering all SAVs at all residue positions predicted as conserved to affect function reached 68.6% (Q2: effect/neutral; for PMD) without optimization, compared to an expert solution such as SNAP2 (Q2=69.8). Combining predicted conservation with BLOSUM62 to obtain variant-specific binary predictions, DMS experiments of four human proteins were predicted better than by SNAP2, and better than by applying the same simplistic approach to conservation taken from ConSeq. Thus, embedding methods have become competitive with methods relying on MSAs for SAV effect prediction at a fraction of the costs in computing/energy. This allowed prediction of SAV effects for the entire human proteome (~20k proteins) within 17 minutes on a single GPU.