You only need to supply the material that has a cost
As you quote
But if a cost is indicated for a component, a character must have that specific component before he or she can cast the spell.
If the intent was to have you be unable to replace all of the components with a focus, it would say "a character must have all the components for the spell" -- but it does not, it just says "that specific component." A spellcasting focus can replace all the other components as normal.
Some spells do require you to have all material components on hand
However, some spells indicate that all of the material components have a cost. In this case, you must provide all of the components. For example, the MC of find familiar is:
10gp worth of charcoal, incense, and herbs that must be consumed by fire in a brass brazier
The charcoal, incense, and herbs must cost 10gp altogether, and they must be consumed by fire in a brass brazier. A spellcasting focus cannot substitute for any of the components here because all the consumable parts also cost money, and they specify the manner in which they must be consumed.
Divination is another example, which requires:
incense and a sacrificial offering appropriate to your religion, together worth at least 25 gp
In essence, the components you may not replace with a spellcasting focus are only those which have a gold cost, but all other materials can be substituted out as normal. However, some spells have material components which all share a cost, such as find familiar and divination. In these cases, you need to have all the components on hand.