[prmph]

Exactly.

Many materials would have compressive strength easily, just by being relatively uncompressible.

But most loads have a (troublesome) tensile component. Fundamentally, the ability of a rigid material to resist deformation (in the most general sense) is what is most important, and that requires tensile strength.

See this comment elsewhere in this sub-thread that explains it probably better than I did: https://news.ycombinator.com/item?id=43904800