A lot depends upon the allergic person's threshold.
With a very LOW reaction threshold, the answer to that question is "no, not always-- sometimes you have to investigate further to know what a company's policy is, and what it means."
With a more moderate threshold-- which, if you've not been seeing reactions and you've been relying upon labels for several months, seems likely
-- then yes, relying upon labels is MOSTLY going to be just fine. You may want to be cautious about new items, and those manufacturers that you're new to/unfamiliar with, and of course follow general cautions about particularly high risk foods (baked goods, deli items, restaurants, etc.).
Why choose the latter approach?
For one thing, it's far far less restrictive.
Secondly, it's far easier to teach OTHERS how to do it.
Thirdly, there is increasing evidence that points to 'excessive' avoidance as a means to depress that triggering dose down into VERY low amounts of the allergen...
The reasons that I would go with that approach unless you see a reason why it needs to change are that eventually, you'll need to turn this over to your child, most likely during adolescence as he starts to gain independence from adults, but even before then as he attends parties, playdates, classes, camp, etc. Your child and other adults without your expertise will need to do whatever it is that you opt for, so basically you want to spend your time now TEACHING how to do it right. It is really a tall order for adolescents and inexperienced adults to just read every label every time, frankly... and anything more than that is likely to lead to occasionally APPALLING lapses, which are super-dangerous.
You can always reevaluate at any time if you are still having problems with that approach (trust the label), after all. This would NOT be good advice for an allergen not included in FALCPA's mandatory labeling, incidentally.