There is methods that does work. But 99% of the time the APU is working and its never an issue.
Newer types, things are setup differently so there is no issue shutting down the left engine. Which is what we do on the A220, sometimes I don't start the APU because I know we can do it safely. But only in certain airports eg Helsinki or Schiphol, would never attempt it in say a Greek island. I just shut down the Left engine while taxing the turn coordinator plugs into the forward coms port and I confirm LH engine off clear attach GPU and dock the air bridge. Reply front chocks in wilco. Then after 60 seconds we get the light come on in the cockpit external power available. Press the button then kill the RH engine then beacon off. And single engine taxi is banned in icing conditions.
To note though there is no requirement for me to have a working APU to do single engine taxi. I know that on other aircraft types you're not allowed to do it with a MEL APU. a220 One engine powered generator online can provide enough power, to power the whole ship and there is backup to power the brakes.
Also that safety cone is idle thrust without ice protection on. Which would have a N1 of about 25% if I have cowl icing on it will go up to 35% and if I am taxing single engine without the APU it will be around 32% without cowl icing on.
With 78 seats I presume it's a EMB135 which is quite a short aircraft and due to being a swept wing jet there will be less room between the pod and the nose connection points than say a Q400 turboprop with 72 seats.
The first hole in slice of cheese is the APU being broken we shall see what the report says about the rest of it.
And you really don't want to be anywhere near the technicians when they are doing engine runs with the cowls up. They crawl all over the things while running and sometimes quiet high power settings. I can't watch and utterly hate being in the cockpit while they are doing it. Thankfully in my current company Pilots are banned from being used for technical engine runs.