The trade being made is an exchange of upside for income, and it is worth stating in those blunt terms. The investor gives up whatever the stock might gain above the strike price during the life of the contract, in exchange for the premium collected today, regardless of what the stock does afterward. If the stock stays flat or falls, the investor keeps the premium and still owns the shares. If the stock rises past the strike, the investor's total return on the shares is capped at the strike price plus the premium already collected, no matter how far above the strike the stock actually goes.
The downside is not covered by this strategy at all, and the name invites the opposite assumption. "Covered" describes the call, meaning the writer already owns what might be delivered; it says nothing about protecting the shares from falling in value. If the stock drops, the investor's loss on the shares is reduced only by the premium already collected, and can otherwise run all the way down with the stock, the same as it would for a shareholder who wrote no option at all. A strategy that actually limits downside is a different combination entirely, buying a protective put, which is described on our page for the put option.
Assignment is the mechanical event that caps the upside, and the buyer, not the writer, controls whether and when it happens. If the stock is above the strike at or before expiration, the option's buyer can exercise it, and the writer is obligated to deliver 100 shares per contract at the strike price, whatever the market price happens to be at that moment. The writer does not choose whether or when this happens; the buyer does. If the stock stays below the strike, the option simply expires and the writer keeps both the shares and the premium, free to write another call against the same position.
The strategy is commonly framed as generating "income," and that framing is accurate about the mechanism and easy to overstate about the outcome. The premium is real cash received up front, which is what earns the income description. But it is compensation for giving up upside, not a free addition to a stock's expected return; a stock that rallies hard past the strike would have earned the holder more by simply holding it uncovered. The strategy tends to look best in a flat or mildly rising market and worst in a sharply rising one, which is the trade-off it is built around rather than a flaw in it.