(Since it looks like I've understood the question, I'll turn the ideas in my comments into an answer.)
As I see it, the students are finding the situation stressful because they are required to do something they feel uncertain about, there is an obvious cost to failing, and they don't feel confident in their ability to meet the required standard.
The proposal is to use students' work, which they are already feeling worried about, for the benefit of others (not themselves). Having been on the receiving end of this themselves (except the first few), they should feel a bit more confident that they understand the requirements. On the other hand, now in addition to the academic assessment they are also to be assessed by their social group, either in their presence or otherwise (some will find one more stressful, others the other). Most PhD students can work out for themselves what aspects of a talk were worth copying, but explicitly pointing them out in a group setting is more judgemental, even if only positive things are directly said ('damning with faint praise'). They will have to spend several more years with these people, providing moral support to each other. The overall effect then is to slightly reduce the likelihood of failure, but significantly increase the cost.
I think the same benefit could easily be gained in a much better way. Choose a student who did a good job in their presentation, and ask whether they would be willing to do a repeat to help others. Some will be too shy, but it shouldn't be long before there is one who is willing. For a repeat performance, there is no academic cost attached. Messing up ultimately won't matter, so the student will feel more confident. They also have the additional confidence from being chosen as an example of a success. This time, they are essentially being honoured within the social grouping, rather than critiqued, so they are being rewarded rather than being put at risk of shame. By videoing the talk, you would only need to go through this once.
The main method I've known for students to become less stressed about having to give a presentation is for them to have opportunities to practice doing so where the potential cost is very low, the main example being in junior seminars (student-run seminars, preferably without faculty present), although reading groups are another avenue. In addition to direct practice, such activities build community within the group, which is likely to reduce the difficulty because students can find out what is required of them by discussing with their peers who have already been through the process. If you'd prefer to have more direct input, one option might be to have a graduate class whose explicit purpose is for everyone to get better at presenting, by having people give presentations that are then discussed (no grades though).
