We can make peptides but not proteins.

The key difference is length. People typically call oligomers of ~50 amino acids peptides. Typical proteins are 100 amino acids or longer.

As others here point out, you can do solid phase synthesis for peptides. However, the limited coupling efficiency (the success rate of adding an amino acid to an existing peptide) and the tendency of growing amino acid chains to adopt conformations that interfere with coupling make it impractical to synthesize proteins.

For example, let's assume that we have a typical coupling efficiency of 98% for adding a single amino acid. For a 50 AA peptide, the success rate would be 0.98^50 = 36%. For a 100 AA protein, this rate drops to 13%, and 200 AA would imply only 1.7% yield.

The above estimate is likely far too optimistic as coupling efficiencies likely decrease with length.

When biological enzymes make protein, they use a molecular machine with error correction to ensure very high coupling efficiencies.

Even if we could make 100 AA length proteins synthetically, another problem is getting the protein to fold correctly. Proteins have their functions because they adopt a specific 3D shape. This 3D shape results from proteins folding in a specific way as they are being synthesized in the biological environment. When we make peptides by chemical synthesis, they are in a non-biological solvent. If you drop a long chain of amino acids made this way into the biological context, they could well fold into a tangled mess that will never function correctly.

RNA is the same way. Biotech industry typically synthesize short RNA drugs (less than 100 nucleotides) with all sorts of chemical modifications using synthetic chemistry), but for long mRNAs such as those in the Covid vaccines, they use enzymes to do in vitro transcription.