The least tolerant part
On this page11
The requirement is phrased about the product as a whole
Care labels must state what regular care is needed for the ordinary use of the product. One product, one set of instructions.
And the warnings paragraph is phrased about harm
If there is any part of the prescribed washing procedure which consumers can reasonably be expected to use that would harm the product or others being washed with it, the label must contain a warning to that effect.
Any part of the procedure, and harm to the product
Neither phrase distinguishes between the fabric and the things attached to it. A garment is one product for the purpose of this paragraph.
From which a straightforward reading follows
If a component attached to a garment would be harmed by hot water, then hot water harms the product, and the first of the five silences is not available. A temperature has to be printed.
This is a reading, and it is offered as one
Nobody on these pages has determined what any particular notion tolerates, and nothing here is advice about how to label a garment. The point is that the rule is written about the whole product and does not carve out hardware.
Which makes the claim implicit in a short label larger than it looks
A bare machine wash, tumble dry asserts the harshest version of five procedures is safe for everything in the garment, including a plated snap, a painted button and an elastic waistband.
The example the rule itself gives is about color, not hardware
If a shirt is not colorfast, its label should state wash with like colors or wash separately. Which shows the mechanism: a property of one part of the product constrains the instruction for all of it.
And there is a matching provision for alternatives
Warnings are not necessary for any procedure that is an alternative to the procedure prescribed on the label. If an instruction says dry flat, the label need not add do not tumble dry.
That paragraph keeps labels short in a different way from the silences
The five silences shorten a label by permitting an omission where the harshest setting is safe. This one shortens it by removing warnings that are redundant against an instruction already given.
Two different economies, both aimed at the same thing
A label small enough to be stitched into a garment and still be read years later, which is what the permanence requirement demands of it.