Unfortunately (La)TeX doesn’t really abide by the rules of stdout and sterr, owing (I assume) to its origins in the early 80s. But there are some switches you can invoke to alter the amount of information being shown.
Execute latex with either the -interaction=nonstopmode or -interaction=batchmode switches for non-halting behaviour even in the case of a syntax error. nonstopmode will print all usual lines, it just won’t stop. batchmode will suppress all but a handful of declarative lines (“this is pdfTeX v3.14…”).
These can also be invoked from within the document with \batchmode and \nonstopmode, but this is less useful for the situation you’re describing.