Why don't 747s use unreliable, but cheap, jet engines instead? I'm a neophyte who was reading a little bit about jet engines, and the economics behind them. In particular, I was curious about what makes jet engines so expensive compared to piston engines.

A recurring theme that came up, again and again, was the cost of quality control, and related factors like product liability. Jet engines are designed to be ultra-reliable under conditions of high stress, and understandably that drastically adds to the cost of manufacture and design.

However, what doesn't make sense is why these standards are so universally high, even in the case of jets on multi-engine craft with redundancy. It would seem that you definitely want reliability on a single-engine craft, a multi-engine that can't fly when it loses an engine, or even a two engine craft.

But for a four-engine aircraft like a 747, why pay so much for ultra-reliability? A 747 can stay at cruising altitude with only 2 engines. Modern jet engines have a failure rate of about 0.01 per 1000 hours. A 747 could use engines with a failure rate of 1 per 1000 hours, and still achieve the same safety profile as an aircraft with only 1 engine of redundancy. (Assuming engine failure rates are independent)

I'd assume that relaxing the jet manufacturing process to allowing 100X less reliability would drastically reduce costs. It seems like we could be manufacturing very cheap four-engine aircraft by allowing for high-failure rate engines. What am I missing here?
The cost of a jet engine is mainly dictated by the operating conditions: high temperatures (sometimes above the melting point of the metals used) and high rotation speeds. This means you need high-performance materials even for a low-reliability engine: if you compromise material performance too much, it'll fail on the first start. 

Failures are risky. Having to shut down an engine mid-flight is one thing, but we've had some recent failures where chunks of metal were launched at high speed into the wing and fuselage. 

The industry has gone the other way: by making the engines more reliable, it's now possible to build a long-range airliner that uses only 2 engines instead of four, reducing the purchase, maintenance and fuel cost. It makes little sense to set up a special B-quality production line for a single aircraft type that's being phased out anyway. 

Failures are costly. An in-flight failure means taking the aircraft out of service to repair the failure, which means having extra aircraft on standby to replace it.