The real reason for this has nothing to do with whether or not AD, Crit, Attack Speed, Move Speed, or Armor Pen are any better than one another on an AD Carry, it is all about the multiplicative nature that these stats exhibit when built together. This means that while every one of these stats are valuable on their own, they each increase each others value when stacked.
AD is great for doing more damage. Each point of AD you buy (not counting resists) gives you an additional point of damage. If you buy 100 base AD and buy 100 more, you are effectively doubling your damage. Attack speed calculation is a little bit convoluted but to simplify the math let's say that buying 100% attack speed doubles your attack speed from .75 to 1.5. Both methods double your damage, and might cost a similar amount of gold too. It gets trickier when it is time to buy a second item!
Continuing with the AD example, we blue pill with more gold and want to increase our damage. AD worked great to double our damage the first time so why not get more? If we buy another 100 AD, we would have 300 now... Cool! It sounds like you would gain the same benefit from the second 100 AD as the first but that is not true. Since this time you had 200 to start, you are only increasing your current damage by 50% instead of 100%! That is HALF the amount of damage gain as you got the first time. Suppose instead you purchase attack speed this time. 100% attack speed will again increase your damage by 100% because you are hitting for 200 damage twice as fast instead of hitting for 300 at the same speed.
TL;DR: Mixing stats gives you more damage than buying just one!
Using this information we actually determine just why AD carries do more damage than any other role in the late game. This is because they have many more stats that multiplicatively contribute to their damage (AD, Crit, Attack Speed, Crit Damage, Armor Pen) as opposed to other roles that rely on only two stats (AP and Cool Down Reduction) to deal more damage.
Now that we know to mix our damage sources, there is still one problem: what do we build first? This matter is decided per champion and per game. I can tell you that Infinity Edge is the most efficient first damage item for Ashe without a shadow of a doubt but there are many more factors than that. You hit on one important one which is safety. Buying Shiv lets you harass from a distance and run away faster. Phantom Dancer has a similar effect. Both let you stack your passive faster. Youmuu's is yet another option, it gives you cool down reduction for your very powerful ult and hawkshot which both can increase your safety and kill potential.
If champion select was highly unfavorable I would probably choose Youmuu's or Shiv so I can use my ult more or poke from a distance. If I had a support with strong peel like Alistair, Thresh, or Zyra I would pick IE so I can bust out the pain without a care in the world. One of the differences that makes an average player great is the ability to change their builds based on the situation and not just what the guide said.