for three weeks (assuming I can summon up the willpower and patience). This should be more than enough time to notice signs of ich.
I think of ich as a 'gateway disease', meaning that it provides a gateway for other diseases to enter. Fish have pretty good immune defenses, particularly the slime coat and the skin. Anything that breaches these first-line defenses, like ich or rough handling leaves the fish vulnerable to other diseases.
The most common diseases, other than ich is probably columnaris (body fungus) and finrot. IME
, these nasties make themselves at home in fish that suffered a particularly bad ich infestation.