I gained throttle response and (admittedly subjectively) a little bit more torque even just with a TRD axleback exhaust (obviously no ECU tuning). I accept I could be imaging it but I don't think I am. If there is any more power though I wouldn't think it more than a few hp at the very most - probably more like 2. But it was more the throttle response that I really liked about that modification.
But for a serious gain I agree with what you say. Here in Australia we have a production Toyota 86 Racing Series and the cars are remarkably similar to showroom stock. They obviously have to have significant modifications related to safety and the brakes are upgraded but so far as the drivetrain is concerned, the only mods are the exhaust (from headers all the way back) plus an ECU upgrade. The intake including air filter is 100% stock. It was explained that the two upgrades (ECU and exhaust) had to go hand in hand and if they didn't, it just wouldn't work at all. The competition cars in this configuration have 179 kw flywheel output 7,500 versus the stock output of 147 kw at 7,000. Torque is also considerably raised lower down - up to 246 nm from 205 nm.
Granted the 1.8 Corolla engine is not the 86 engine but it is still a normally aspirated current model Toyota and I think the same would apply in that to get the maximum out of a new exhaust, an ECU tune is necessary.