The next stage will be adding speed, the next stage adding strength, the next using combos, and after that some friendly sparring, which we tried actually, but I was simply overwhelmed that I am not "allowed" to stop and think after parrying 4-5 punches but there are dozens and dozens more coming.

You're not feeling overwhelmed because sparring requires a level of skill that you haven't built yet. You're feeling overwhelmed because you wandered into a completely different skill set. The only way to get comfortable with sparring is to do a lot of sparring -- although partnered kata can be helpful in drilling responses to specific situations, and a lot of weapon and grappling arts spend most of their time doing that.

This actually touches on one of the big problems in martial arts pedagogy: there's a large body of skill that doesn't come without freeform competition, and you can't simply ignore that without throwing out application entirely; but the earlier you start sparring seriously, the more likely it is to build in bad habits or to lead to injury. Martial arts' approaches to squaring that circle often define their styles more than their actual technique does; you'll find almost all aikido techniques in jujitsu, for example, but their pedagogical approaches couldn't be more different.