The general rule today is that wider is usually better, up to the point where it exceeds your frame's and rims' limits, taking you into a different bike type.
Your photo shows what would be considered a gravel road by Western cyclist standards. Such roads are best served by 32 to 47mm tires, on the wide side of road tires, what's known as gravel bikes today. 42mm will be better than 35.
Tire rolling resistance, as measured on a flat surface, is of most concern on good quality asphalt/concrete roads. Elsewhere, the better traction provided by wider tires, and especially tires with some thread, wins hands down. Narrow tires are vulnerable to surface irregularities and don't save much except for weight and drag.
Since bikes that are "just right" might be expensive, if cost's a concern, you could also consider a mountain or hybrid bike, which tend to use ~2" wide tires. They're not as well-geared for speed. The main reason 2" tires are "slower" than 47mm is that they're usually fitted to "slower" bikes, and tend to be knobbier.
A hardtail can make the ride easy on your hands, eating up all the bumps, but a suspension fork is still very optional here. Against small potholes, 2" tires at moderate pressure are sufficient suspension by themselves.
Wider than 2" is off-road tire territory, and any increase over 2" won't add value here. I often have to ride asphalt (to the actual trails) on a 3" tire, and you begin to feel the extra weight and rolling resistance there. Wider than 3" is fatbikes, and these are for snow or very loose terrain. With 35-47mm, the loss vs narrow tires is just a small amount of weight and drag.
At your level, the "engine" is more likely to limit the speed than the bike. The differences comes from road, CX or gravel bikes having faster gears and a more aggressive riding position, which saves on drag. But this position also takes some effort to get used to.
All in all, it's not SUV vs racecar; it's maybe 1-2 km/h of average speed between a "fast" cyclocross and a "slow" mountain/hybrid bike, for a rider that's just starting out and doesn't take either to their limit. Pick the bike you're comfortable with, both riding and maintaining, not the one you think might be faster.