Are my fears of the anchor pieces popping out justified?
Yes. This is an especially big concern when the climber has already placed the first piece of pro above the anchor, but falls before getting a second piece in. The fall factor can be large, and the direction of pull is up. If you don't have any gear that can hold against an upward pull, then your anchor isn't an anchor in this situation.
Additionally, there is very rarely an option to place a nut to protect against an upward force.
My experience is fairly limited, but in my experience it's not true that there is very rarely such an option. There should be, and there usually is. If you're on a well known rock-climbing route, then the route should have belay stations where it is possible to place adequate protection. If the belays were inherently unsafe, then hopefully nobody would have published a route description that told you to use those belays.
It may take some ingenuity to find an appropriate placement. It may be more difficult to do with only passive gear, although in long-established areas like Tahquitz, the routes predate SLCDs, so it should be possible to slot a nut somewhere. Take some time and look around for placements. In recent memory, I recall a lot of cases where the piece that protected against an upward pull was a small cam.
If you're using passive gear, then you probably need to place two pieces in opposition, one above the other. Then you tension them against each other, e.g., with clove hitches on a sling.
This is the kind of thing that is discussed in great detail in Long and Gaines, Climbing Anchors: http://www.amazon.com/Climbing-Anchors-How-Climb-Series/dp/0762782072