Intercepting flash orders is front running. I don't see how you could even be confused about this issue. That is why the SEC wants to ban it, and some exchanges have removed the feature. This corrupt business practice creates artificial information disparity which wouldn't normally exist.

I'm undecided about whether I think front running is fair

This is a red flag that no one here should be asking you for moral advice.

Front running: Joe wants to buy shares. I buy ahead of Joe, driving up the price, let him buy, then I sell my shares, profiting off the price delta. This costs Joe money, since he buys at a high price and sells at a lower price.

Flash trading: Joe wants to buy shares at price 10 or better and places an order on NYSE. The best ask on ARCA is 9.99, but the best ask on NYSE is 10. NYSE gives me the option of filling Joe's order at price 9.99 (rather than routing the trade to ARCA), saving Joe the cost of routing.

Flash trading and front running are just not the same thing. Flash trading only happens to traders who chose for their orders to be flashed. All flash trading does is moves the trade from ARCA to NYSE.

Flash trading: Joe wants to buy shares at price 10 or better. You are Goldman Sachs, you see Joe's order before anyone else.

You see shares costing 9.80, so you buy it up quickly, and sell it to Joe for 10, making a profit of 0.20 per share while driving up prices for Joe.

That is front running.

You apparently do not know what a flash trade is.

A flash trade gives Goldman the opportunity to fill Joe's order at the NBBO price before it is routed to another exchange. It does absolutely nothing else. The person receiving the flash is even prohibited from making offers on that security on other exchanges for a few milliseconds after receiving the flash.

This gives Goldman an advantage over other high frequency traders since it gives Goldman a higher fill rate, which is definitely unfair.

Apparently you don't. The allure behind flash trading is to avoid rule 602 in regulation NMS: you are not required to fill the order at the NBBO. You can use dark pools to fill orders.

Yes, but the order would not be filled at the dark pool price without the flash trade. Joe's order would be filled at the NBBO on another exchange (NOT the darkpool) and Joe would pay an extra routing fee.

If Joe wanted to fill the order himself on a darkpool, he would not have asked the exchange to flash his order.