Update - this has been bugging me for a while, I finally had to write it out. It's speculation (not taken from the scripts), but internally consistent with at least the first two movies.
In regards to <PERSON>'s question, "why would Skynet invent a time machine that only works on living tissue" - I'm not convinced that a) Skynet did the inventing, or that b) it only works on living tissue.
First, we don't know that Skynet really invented time travel - even if one of the characters said <PERSON> did. It's entirely probable that some (human) laboratory had worked out the basic principles, but alas, never got a chance to publish their exciting results, prior to most of the human race being wiped out in a nuclear Armageddon.
Second, the Temporal Transfer Field doesn't only work on living tissue - see the T-1000 for a counter-example. In fact, the problem that <PERSON> encountered, was a lack of understanding of the principle of Chrono-topology.
Chrono-topology is the principle that the energy required to activate the Temporal Transfer Field, is correlated to the number of bundled timelines inside the Field 1 . One timeline exists for every object that had a separate existence through the past.
1: Note that power requirements appear to be 1.21 Gigawatts per timeline, making it prohibitive to send more than a single object.
For instance, a (naked) human has one timeline, as he exists as a unitary object going back in time. This includes his hair/fingernails, which are biologically dead tissue, because they are all in the same timeline. If the field only worked on purely living tissue, the subject's hair (and fingernails - ouch) would need to be removed.
If the subject human puts on a pair of jeans - now there are two timelines inside the field, as the jeans have their own existence and path back through the past. The same for picking up a gun (and the bullets).
The T-800s, having been created and (presumably) immediately had skin grown on, have a single timeline. Likewise for the T-1000s. 837
If they were to pick up a weapon, that would double the number of timelines in the Field.
Why the increased energy requirements? The TTF, when activated, is basically trying to push an object out of its existing temporal path back in time, into a different path. The more distinct timelines, the more energy is required for the Field. A timeline is a four-dimensional path made by a physical object, travelling forward through time. Think of a bikepath through soft dirt, where the wheels have made a trench. If you ride a bike back down the trench, it takes an effort to force the wheels out of the trench.
Now, it is true that Skynet could do a number of things to circumvent this restriction, but see the next section for an explanation of why it doesn't.
Skynet is Pac-man on steroids.
Skynet is an AI, and doesn't process scenarios like a human. It follows certain rules programmed into it, no differently from one of the ghosts in Pac-man (just slightly more complex).
Two rules in particular, Local optimum, and Satisficing, prevent Skynet from doing more than the minimum planning necessary. First, when considering the sub-goal of how to send a cyborg through a time-field that seemed to only work on flesh-covered objects, Skynet would run through its list of available assets, realize that it already had flesh-covered cyborgs, and move to the next step - how to arm the newly-arrived Terminator.
For this second sub-goal, Skynet would realize that the Terminator could locally acquire weapons, sufficient to take out a waitress. Having satisfied both of its subgoals, Skynet would stop evaluating additional options, and carry out its plan. It has achieved a local optimum, and doesn't realize (or care) that there might be a better answer elsewhere.
Skynet, as an AI, just isn't creative, and doesn't seem programmed to make intuitive leaps, which would be obvious to a human. It processes trees of (known) options, and picks the ones that meet the given criteria.
Lucky for us... 64