For some reason in F1 the combination of refuelling and tyre stops never really worked that well form the point of view of making races unpredictable. In most cases it meant that out laps would be slower than in laps, so the trick was to stretch your fuel, and overtake by pitting later. You could go for a lighter load and more stops, but unless you were a lot faster and could make the passes on track it wouldn't work well. At least with the undercut in today's races, you are almost certain of gaining position by pitting sooner, but it might come back to the other guy later because his tyres are fresher.
In Indycar they have refuelling and tyre changes, but I think there are two factors that work to make the races more interesting/chaotic. Safety car periods are more common, so it's generally worth gambling on pitting now rather than sticking to your original plan, because it might pay off later. In F1 you'd pit under the SC, but simply adjust your strategy to the end from there. The other thing is the limited fuel tank size, so there's a maximum number of laps that everyone can do. But this leads to fuel saving. If you think fuel saving in F1 is common, then you haven't seen the guys in Indycar gambling on a 2-stop strategy and trying to make it to the end in the last stint.
Would you want to turn F1 into Indy-style racing like that? I'd rather they kept their current characteristics.