The APU is an essential tool, but hardly necessary. APU's on different airplanes accomplish different roles. All provide bleed air for air conditioning and other purposes, most generally drive at least one electrical generator, and some also run hydraulics, too. However, each of those functions can also be accomplished by outside sources and equipment, until the aircraft's engines are running and supplying bleed air, electrical power, and other aircraft system's needs.
I've flown airplanes around the world with an inoperative APU. It's an inconvenience at times, requiring keeping an engine running on arrival until ground power is attached, or cross-bleed starts and slow starts from ground carts...but it's do-able.
From a passenger perspective, not having air in the cabin can be uncomfortable, especially in a full cabin on a hot day. An airline may not simply have another one hundred million dollar airplane on hand to substitute for you...after all, how many spare cars do you keep handy in case yours breaks, or how many spare homes do you own in the event you might lose one in a fire or flood? A spare isn't always available or convenient...or may be in use for a similiar situation affecting another airplane.
Each airplane has a minimum equipment list which specifies what can be inoperative, for how long it may be that way, and under what conditions the aircraft can be operated, as well as specific maintenance or pilot operational requirements to make the flight safe and legal. The APU will be included in this list.
We can't start our APU in flight anyway; it's a ground function item only...so not having one really has no impact on our ability to go fly. It might have an impact on whether we can start the engines, keep the airplane warm or cool, or do certain things on the ground, but it's perfectly legal and those are operational convenience issues, rather than things which affect the safety of flight or the legality thereof.