A flat map has to make a trade
A familiar outline can feel like neutral geography. It is still the result of a mathematical choice. Transferring a curved surface to a plane cannot preserve every relevant property everywhere, so the right map depends on what a reader intends to do with it.
The U.S. Geological Survey explains that projections have different uses. Mercator maps strongly enlarge areas near the poles. Equal-area projections preserve area relationships while altering shapes, and equidistant projections preserve distance only along specified lines. The phrase 'accurate map' needs an accompanying question: accurate for which property, over which area?
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Scale changes the practical problem
USGS also notes that a properly oriented Transverse Mercator map of a small area can avoid most of the conspicuous distortion associated with a world Mercator map. Its working manual documents projection choices used for different mapping purposes. A criticism of one world map should not be applied blindly to every local map whose projection name includes Mercator.
For an area comparison, choose an appropriate equal-area representation or calculate areas with a suitable geospatial method. For a route or distance question, identify the permitted measurement and coordinate system. Do not measure an arbitrary line on a screen and assume the displayed scale applies equally throughout the map.
A caption can prevent a false conclusion
Keep the projection, geographic extent, scale and data date beside a map image so another reader can check it. These details matter when interpreting an island's apparent size or the closeness of two coastlines. Before drawing a conclusion from the picture, establish how it was made and which measurements it supports.