Flight instructor on training flight with student reports a NMAC and other questionable actions by another aircraft also on a training flight.

Narrative

Aircraft Y is a flight school aircraft operated by Flight School A under part 61. Aircraft X is a flight school aircraft operated by Flight School B under part 141 and 61. I am a flight instructor with Flight School B. Both aircraft are based at ZZZ and departed ZZZ around XA:30 local. Upon reaching the run-up pad; Aircraft X noted that Aircraft Y's engine was not running; both pilots were distraught with their door open; talking on the phone to call for help; and attempting to start the engine. ATC asked if they still had an issue and they stated on frequency that it was a 'parking brake issue.' After 15 minutes of attempting to restart the engine; Aircraft Y's engine began running as we taxied to the runway to depart on an IFR flight plan. To our surprise; Aircraft Y did not taxi back to the ramp for mechanical inspection; but rather taxied to the runway with a cold engine that had died in the run-up area and been unable to start for an extensive amount of time before their departure.The conditions were VMC with mixed changing conditions and areas of IMC. Both aircraft had two occupants. Aircraft Y's second occupant was presumably a flight instructor. Upon reaching our first airport to perform an approach; ATC informs us of VFR traffic rapidly converging on our course; and that he was not talking to them. We advised we would look for the traffic. ATC clears us for the approach after making visual contact with the traffic. At this time; Aircraft X is established on the final approach for the ILS XX at ZZZ1. I notice that Aircraft Y appears to also be establishing themselves for the same approach and is rapidly converging on our position. I make multiple calls on CTAF to the aircraft and ask if they are on frequency and to say intentions. There were no responses. I ensured visual separation but noted that Aircraft Y was oblivious and ignorant to maintaining VFR on an instrument approach they had no clearance for. After the instructor of Aircraft Y leads his student to a near-collision under VFR; the student finally responds to my inquiries on frequency. The student of Aircraft Y makes an admission on frequency to confirm that they did have an engine failure in the run-up area at ZZZ and took off subsequently.The actions taken by the instructor of Aircraft Y have no place in general aviation. Taking off immediately after an engine failure with no mechanical troubleshooting followed by a blatant disregard for human life is simply unacceptable behavior. Without my evasive actions; this instructor would have very well led his student to a collision.