Why can’t aircraft use GPS to generate the IAS, instead of using pitot tubes?
To answer the other part of your question, in a scenario where you lose all 3 speed indications, you are left with pitch and power.  If you know the airplane really well and are sufficiently skilled at flying, this should be enough.  

For example if you've been flying some jet for a while you will know that at Vref on a 3 degree glide slope flaps and gear down you will have an fan RPM range of say 60% at lighter weights and 65% at max landing weight.  So when on the glide slope you just hold the pitch attitude that keeps you on slope and keep the power there and the speed will take care of itself, within reason.

Same if you're just flying around in level flight.  Just holding the pitch attitude and setting power at the value you're used to will keep your speed somewhere in the ballpark.  In fact, at the extremes, it's quite easy to tell the difference between 250 kt and 330 kt just from the sound (the roar at Vmo is unmistakable).  

On a descent at idle, you'll know from experience that the pitch attitude is, say, 15 degrees below the horizon at 200 kt.   So you just fly the attitudes you know will work, with power that's appropriate to the speed range you want, and you'll be close enough not to fall out of the sky.

On top of that, the manufacturers all provide pitch/power setting tables in their QRH procedures that can be used to maintain speeds close enough to avoid falling from the sky (fallout from some of the clear-air pitot icing incidents). 

I know personally of an Atlantic Southeast Airlines CRJ crew that lost all 3 airspeed inputs from ice crystals, used their pitch/power skills to manage things on decent until they got their speed indications back when they were lower.  But they would have had no problem landing the plane with no airspeed if it came to that, and that was before there were QRH pitch/power tables.

The Aeroperu guys were unfortunate in that they went into brain-lock and became disoriented in the darkness.