Here is how I think we could have flagging and voting back, without compromising the philosophy behind the new review tasks:
The picture shows an answer on the Low Quality Posts queue, but the question below it could have the same link, so we could also vote on it or flag it. Maybe the new link should only be available to users with a certain reputation, or who reviewed at least so many posts.
- Current UI is almost unchanged until you click the new link.
- New link does not draw too much attention, and focus is maintained on the main task and its suggested actions.
- Showing the voting and flagging links still requires an additional click (good for extra tought on your actions), but doesn't require a new browser tab (ajax assumed).
- Implementation shouldn't be too difficult, since the server-side code for this actions obviously exists, and client-side code could be adapted from the old review system, the 10k flag queue, or elsewhere.
A concern with allowing voting in the Low Quality queue specifically is that you're not viewing answers in the context of other answers, or a question in the context of its answers. You're never able to vote in these circumstances normally - it's worth remembering that voting directly from the list of questions was considered early on, and discarded as too likely to produce bad results.
(But I think "you're never able to vote in these circumstances normally" is a little too strong; sometimes there is no need for more context, and it's not too unfrequent.)