It's natural -- responsible, even -- to consider the "what if" any time you're introducing a new dependency on a service. You may not consider it consciously, but even the most reliable services we use fail us sometimes. It would seem impossible not to develop this sense at some level.
> Conventional wisdom says that reliable services that are always online would be perceived as being more trustworthy than those which go down.
That's not conventional wisdom, it's naivety. 100% reliability doesn't exist over any significant time scale, thus the assertion of this particular "conventional wisdom" is a wisdom that no one has developed. If it is anything, it is a conventional misconception. We might tell ourselves this with our consious minds, but we are all weary of the risk of failure at some level.