Lee, I don’t think we’re on the same wavelength yet. I think to drive as straight off a corner as possible is the best solution for a high performance two stroke, but not so much for a rental four stroke. In a high performance kart you may sacrifice cornering speed for maximum traction and acceleration, but since a rental kart is so low on power, you can drive the corner much more round in order to keep the speed up. Although, there might be situations when you want a straight exit line, for instance when you still want to hit the apex after a very late entry (common in wet). But in many cases on slicks the outside line is the way to go.
The way to go fast is to drive close to the limit. I am not saying anything new here, but if you are not close to the limit, the kart will just not properly react to your inputs. So you really need to hussle the kart at the entry, like TJ described, otherwise the kart will not rotate from straight. Right after the turn in, you can apply some throttle, just for some minor wheelspin (you can hear the pitch of the engine is quite stable). The wheelspin is just enough to maintain the rotation you created with the big steering input at the entry. So after the entry, the steering input is close to minimal.
I think the exhaustion you mentioned earlier is the result of not driving the kart close to the limit. You must have experienced massive understeer, which is indeed exhausting. So unfortunately I reckon you need ‘the fancy stuff’ but I think you can go much faster with less fatigue!