That's not strictly true. My PhD project has looked at an entire class of peptides that are known to fold in to alpha-helices when bound to a cell membrane e.g. https://www.rcsb.org/structure/2k9b.
There are many others that are known to fold into beta-sheets too.
Ok, so I'm playing with definitions here a bit. Clearly 2k9b adopts a helical structure so it is 'folded' in some sense. However it has no tertiary structure.
Structural proteins like collagen are different again. They have a simple fold that could be described as quaternary (multichain), although that is again an abuse of terminology.
There are also some small proteins (peptides?) held together by disulphide links that have no secondary strucure to speak of. I think these are Class 4 in CATH, but I do not remember.
What is an example of a beta-sheet peptide? Amyloid is beta, I thought, but sheets are not normally stable outside sandwiches, barrels, etc
WW domains[1] fold stably into a three-stranded sheet. Also, not strictly a sheet, but beta hairpin motifs (e.g. tryptophan zippers[2]) are known to be stable isolated from their parental domains.