Switch full-cluster-restart to new style Requests (#32140)

In #29623 we added `Request` object flavored requests to the low level
REST client and in #30315 we deprecated the old `performRequest`s. This
changes all calls in the `qa/full-cluster-restart` project to use the new
versions.

It also fixes a small bug in the test for explain on the `_all` field
that was causing it to not properly invoke `_explain`.

Author: nik9000