There is a distinction between welfare and output in economics. Gross Domestic Product (GDP) is a measure of how much an economy produces in a given period of time. This can typically be done by looking at aggregate expenditures, income, or value-add throughout the production process. It does not measure welfare or well-being, and by definition, is not supposed to.
Welfare through a consumer’s lens is often proxied by estimates of consumer surplus. Consumer surplus is the difference between the maximum amount someone is willing to pay for something and the amount they actually pay. Let’s say food prices go down (and let’s say everyone has inelastic demand for food!), then consumer surplus increases. Assuming that everyone still consumes the same amount of food, there is no change in how much food is produced or consumed, but people are better off. So, we have no change in output, but an increase in consumer welfare.
In this example, the change in GDP could be lower than the change in consumer welfare. It does not understate it, though, because it is not intended to measure welfare to begin with – only output.
Finally, well-being is an even broader concept that includes subjective views on how well someone believes she is doing in life. More specifically, subjective well-being has a lot to do with how one perceives his or her quality of life to be improving, and whether it will improve in the future. Other indicators include confidence in public services, health and mortality rates, job security, etc. These are all outside of the scope of the Gross Domestic Product. Check out The Economics of Well-Being for more.
Therefore, to get a sense of what the level or changes of well-being might be in a country, one needs to look beyond GDP. If you are interested in overall output growth (often used as a measure of total income growth), then GDP is your measure. Though, with the increasing role of information and digital services, that, too, needs a re-think (see Beyond GDP: How Our Current Metrics Mismeasure The Digital Economy).