I use a round pen in my training program, but I don't go and make an introduction about it. I bring them in, turn the horse loose and go to work. Any area I train in, whether it is in the round pen, the arena, the trail etc is pretty equal parts rest and work. However i'm careful about where I rest my horses in any area I work them in to balance things out. If I have a buddy sour horse, the side of the round pen closest to the barn i'm going to work them a bit harder and rest them on the other side facing away from that buddy. If I have a horse that doesn't like you being on the right side I love a round pen because thy get used to you. I'll work them a bit harder out of the left eye and offer a rest when i've turned the horse and i'm in their right eye. I don't use the word punishment personally, but none of my horses think that the round pen is a place for that. Again it's all about balance. I want the round pen to be a good place so I rest a good amount in there. However I don't want being near a buddy to be thought of as a nice place to be so i'll offer the horse to choice to go there but make it a difficult choice and being away from the buddy easy. Making a training area be associated with something negative would be incredibly counter productive.
The reason a round pen is an easy place to start training is that you are not going to tire out the horse or yourself while teaching new things. A 60 foot diameter round pen is plenty of room for a horse to lope a circle but small enough that I can back up a suggestion. Even in the Houston summers here in Texas I rarely work a horse into a sweat (unless they're fat as a hippo, but any horse in decent shape isn't going to be sweaty) because it's not the point. I'm teaching the horse to move off pressure and for those first few sessions it needs to be to the point. If I worked past the point of sweating in those few sessions it wouldn't be easy for the horse, at that point what I wanted (moving forward) would be difficult not easy. I personally don't teach a horse to come in toward me for a few sessions. I want the horse moving forward off of just a point to the next gait, turning into me for direction changes, stopping to face me for a rest and disengaging the hindquarters when I walk toward their butt. Only after all of that is excellent do I get the horse to draw in towards me at all, because at this point i've got a much more respectful horse than I started with and even if the horse DOES try and get pushy I can easily send them off again. Again, the horse isn't tired and needing a rest and having to face me to get it. Each step leading up to this has prepared the horse to do as I ask and the horse always has a choice. If I got a horse so winded it had to come in, it wouldn't learn the lesson nearly as well because the choice was taken away.
The round pen should not cause any anxiety, if a person is doing something to cause anxiety in their horse about being in the roundpen it's going to show up other places as well.