I too would echo the posters above. In the same vein, don't judge a student's skill from a single circuit. Everyone has peaks & troughs - expecially at an early learning stage and often in the same lesson. Consistency for a few laps around the aerodrome is more important to determine if you think someone should be in the sky on their own. You're trying to get a broad picture of their skill, judgement & ability to recognise their own errors and then correct those errors.
Don't forget it's not perfection you're looking for ie like your own skills
, but 'adequate for the task and safe'.
As for emergencies/non-normals I wouldn't send someone solo unless they could handle the sorts of things that would be a worry. The standard I looked for was one that would either get them back on the ground with a good chance of living, or for less severe events, with the themselves & the a/c intact.
For an EFATO ie not appropriate to glide/turn back then at a minimum I wanted them set something close to the glide attitude & head for a clear area with sufficient control to hit the ground right side up. If time & height allowed then check C/heat, Mixt, Fuel selector & pump (if applicable) and trim.
For a failure where a glide back to the field is reasonable then an ability to get land back on the runway from a glide with an overshoot corrected with track adjustment, flap & slip as necessary. As workload permits then the usual C/Ht, Mixt, Fuel & Pump.
If a flapped a/c then the ability to land flapless or part-flapped without damaging the a/c even if not the smoothest.
Depending on type then maybe also a door or window opening. No great expectation other than to ignore the bloody thing & concentrate on getting around the circuit to a landing with no attempt to close the door/window until at least at a safe height & under control.
At a towered field then also radio failure + knowledge of the in-flight light signals.
Lastly, a safe go-round: Add full power, control the pitch to maintain Vy, flap up in stages & trimmed correctly followed by a normal circuit.
I always preferred to have flown with the student in the lessons leading up to a first solo so I could get a better understanding of their ability and as a way to check the non-normals above without necessarily having to do them all in a single pre-solo check. If that wasn't possible then I'd discuss some of them in the circuit during the 3 or 4 laps.