Self-reducibility of Clique
The Repeat loop is not strictly necessary.  The algorithm would still be correct without it.  In fact, the Repeat loop actually has no effect, though proving this takes a little bit of work.

If the graph $G$ has no $K$-clique, then this procedure correctly halts.  So assume the graph $G$ has at least one $K$-clique.  The interesting case is where it has multiple $K$-cliques.

Associate each $K$-clique with a vector $(x_1,x_2,\dots,x_K)$ representing the vertices of the clique, in increasing order.  Now order these cliques by lexicographic order on the vectors.  Let's focus on the $K$-clique whose corresponding vector is lexicographically largest; call this the "golden" $K$-clique.  It is possible to prove that For loop removes everything but the "golden" $K$-clique: when the For loop terminates, all that remains is a subgraph containing exactly $K$ vertices, namely, the golden $K$-clique.  Therefore, the Repeat loop does nothing.

In other words, you can remove the Repeat loop, and this procedure will always produce the same output.  (In particular, you were worired that the Repeat loop might remove a vertex from an actual clique -- but the above reasoning shows that this can never happen.)

So, the Repeat loop is unnecessary.  Why is it there?  Maybe to make the proof of correctness for this algorithm a bit easier.  But if you find the algorithm easier to understand if you omit the Repeat loop, go ahead and pretend it's been removed -- that will work, too.