That a married Jewish woman covers her hair in public is common ground across Orthodoxy. What satisfies the obligation is not, and the disagreement is genuine, sourced on both sides, and does not run along a scale from lenient to strict.
The case for the wig. A sheitel covers the woman's own hair completely, which is what the obligation requires. It is durable, it does not slip, and it allows a woman to meet the requirement without being marked out in a workplace or a street. Ashkenazi authorities from the eighteenth century onward permitted it, and in much of the Ashkenazi world it became the norm.
The case against it. A covering indistinguishable from uncovered hair defeats the purpose of covering. If the point is that a married woman's hair is not on display, a wig that looks like better hair than she had achieves the opposite. Most Sephardi authorities take this view, as do parts of the Hasidic world, and their communities use a scarf, or a scarf over a partial wig, or in some Hasidic groups a scarf over a shaved head.
Reading these as degrees of observance gets it backwards. They are different answers to the same question, and which one a woman follows generally indicates whose rulings her community accepts.
The hair itself
A separate problem attaches to supply. A great deal of human hair for wigs has come from South Indian temples, where devotees offer their hair in tonsure and the temples sell it. In 2004 several Israeli authorities ruled that such hair might constitute benefit derived from idolatrous worship, which would make it forbidden. Sheitels were burned publicly, and a market of some size was disrupted for months.
The ruling was not universally accepted, others held that the tonsure is an offering of the hair rather than to an idol, and that the sale is a subsequent commercial act, and the sourcing question has never fully gone away. It is a good illustration of how far the practical consequences of a halakhic question can reach, and of how tangled two unrelated religions can become through a supply chain neither chose.