Replace my 28 x 1-5/8 x 1-3/8 tires with beefier ones for flat gravel trails

Tire sizing is a very funny thing. The size you list could be on one of two different rim sizes, either 622 or 635 (https://www.sheldonbrown.com/tire-sizing.html). The tire should have another marking on it after the size you listed, something like 622-32. That tells you the actual rim size. Hopefully it's 622 as that is much more common. You may be able to fit a wider tire, but that depends on the design of the bike frame. You can look to see the spots on the frame where the tire comes closest to rubbing. If there is extra room, you should be able to fit a wider tire.