A very logic assessment and I suspect you're right. In the olden days, when we didn't have smart current and voltage control systems, a fast spinning starter suggested a good battery, period. Now it might indeed mean that there's enough to spin the engine, but clever systems are not able to fire the ECU and other vital organs as well.
If you do get to change battery, go for a bigger one if possible, as it will always do better than a smaller unit. I went up a size on the Bosch Silver jobbie I bought for mine a couple of years ago and it's never run up a sweat since. Still fits in the tray OK too, althouh I did need to make up a slightly different securing strap.