How realistic is the aviation crisis in the recent Spider-Man movie?
This depends on the aircraft type, the flight rules and weather conditions as contributing factors.

Larger aircraft or medium sized aircraft operating under general aviation or commercial aviation will in many cases be equipped with a technology called Traffic Collision Avoidance System (TCAS) (See: How does TCAS work?), which can instruct both aircraft to take evasive action if a collision is imminent. This is especially important in Instrument Meteorological Conditions, where pilots do not rely on outside visibility, but on instrument flight only operating under Instrument Flight Rules.

In Visual Meteorological Conditions, aircraft would likely be able to see each other and take evasive manueouvers without ATC or TCAS having to warn them, e.g. this is the case for smaller aircraft operating under Visual Flight Rules.

Aircraft receive clearances and instructions to occupy a certain flight level or altitude, sometimes coupled with clearances for approaches or arrival/departure procedures. If pilots do not receive further instructions, they will start to hold over their last clearance limit and commence lost radio/comms procedures, which include holding for a given amount of time and then beginning the descent towards the airport on defined and published approaches. If the power outage is long enough, the airspace is quite busy and all other frequencies are also unavailable, one of the being the distress or guard frequency, there is a probability that two aircraft on the same level will reach their clearance limit at some point and enter a hold at the same flight level, which could result in a collision.

The scenario however assumes that really all failsafes and contingency procedures have failed.

See also ths related question: What happens when an airport shuts down?