Can a detector fail to detect a wave of electromagnetic radiation because the wave's amplitude is at or near its minimum?

A very brief wave would be described by a wave packet, and if you look the animation in that Wikipedia link, you’ll see that as a wave packet propagates past a location, every part of it passes through that location. That is, the detector cannot see only node,* because at the next instance, the detector would see the peak. Another way to think about this is through the time-averaged Poynting vector ($Releft[Etimes H^*right]$), which is non-zero in your scenario, so the photodetector will measure the wave. If the wave really was nothing but node, then it would be no wave at all.

*”Nothing but node” could occur at a detector if the detector is in a standing wave, but this is a different situation from your wave packet scenario.

From a radio engineer's perspective, a travelling wave sweeps through a fixed point in space and produces an oscilliatory response at that point. The response will repeatedly cross zero in a periodic manner and certainly if you sample only at those time instants, no response will be detected. There is a related argument in the use of synchronous detection. If the local carrier is in quadrature (90 degrees out of phase) to the incoming signal, then there will be no response.

Another point to be made for electromagnetic radiation is that small detectors are usually designed to detect the electric or magnetic field component alone. If they are oriented perpendicular that particular field component, they will detect nothing.

For a standing wave between two reflecting surfaces, there are points in space where the electric or magnetic field amplitude is always zero and a detector looking for that field component will detect nothing irrespective of its orientation.

These arguments pertain to narrowband radio systems. It's not so clear what happens for very short (wideband) pulses or if the quantum properties (photons) are important.

There may be a misunderstanding lurking behind this question.

An electromagnetic wave is not like an interference pattern, in which the power density varies between some maximum and zero. The power density is actually proportional to the square of the amplitude, which is constant (for a relatively long uniform wave train). The "oscillation" of the wave is not a variation in its power density; it a variation in the phase of the wave's components. An interference pattern involves two or more superimposed waves which can result in a temporal and/or spatial variation in power density.

That said, a wave packet is essentially a bunch of waves of slightly different wavelengths traveling together and interfering temporally with each other. Typically, it is a wave train that begins with low amplitude, grows to a maximum, then decreases to a low amplitude. If a detection setup is arranged so that it only has an opportunity to detect the wave packet before or after its beginning or end, it will detect nothing. If the wave packet happens to have a double hump with a zero-amplitude portion between the humps, then a setup arranged so that it only has the opportunity to detect the packet at that zero-amplitude point will detect nothing.

Making a wave packet with a double hump is not particularly difficult. A setup that can only detect the packet during a brief interval covering a few wave cycles is a bit more complicated, but it can be done.

So, the answer is a qualified "No".