We recently had a professional build our kitchen cabinets, and noticed after we paid them in full that the shelves inside the upper cabinets were only 5/8" thick. We applied pressure to the middle back area of the shelves and they flexed downward. Shouldn't shelves be 3/4" thick for rigidity and strength?
TLDR: 5/8" plywood shelves spanning 30" are fine. They aren't going to fall.
5/8" shelves might seem a little slim, especially when you've been primed to think that 3/4" are better, but under normal loads, these should be fine. (Assuming plywood -- particle core is a different matter, and you can probably see what the actual core is by looking at the back of the shelf.)
They aren't going to flex themselves off the shelf pins and cascade down.
If you end up storing your collection of osmium or tungsten up there, then the shelf might sag a little and somebody will install a 1x2 of hardwood, oriented vertically, at the back of the shelf and that problem will go away.
edit: just to add mention of a very useful tool: the Sagulator that says a 30" wide, 12" deep, 5/8" ply shelf with 30# per foot uniformly distributed will sag .02" (<2mm) or less.