If the stock price ends up below the strike price, you've lost the premium. But because, in your hypothetical, the strike price plus the premium is equal to the current stock price, that means that if the stock price at expiry is above the strike price, it's the same as if you had just bought the stock. That is, you can pay the premium, and then pay the strike price, and not pay any more than if you had just bought the stock at its original price.
What this means is that you are getting the full benefit of buying the stock at its current price (if the stock goes up $X, the profit from buying the option will be the full $X), but less of the downside (the most you can lose is the premium).
This means that if you buy one of these options and then short sell the stock, there are situations where you'll make money (if the stock craters, your short position will make you more money than the option loses) and no situation where you lose money. This is known as "arbitrage", and one of the principles of economics is that with an efficient marker, arbitrage cannot persist. If a situation you describe were to exist, traders would do the combination described above (short the stock, buy the option), which would drive down the cost of the stock and drive up the cost of the option, until (premium+strike price) is significantly larger than stock price.
There's been mention of there not being any time value, and you've expressed confusion because in your hypothetical there is time between the purchase and the expiry, but what they're getting at is this lack of difference between (strike+premium) versus current price.
For the reasons I've given above, in the real world the strike price of an option plus its premium is higher than the price of the underlying. The amount of this difference reflects the uncertainty in future stock prices. Since options are hedges against volatility, the more uncertainty there is, the more it costs to insure against that volatility. So more volatility generally means higher option premiums. Since a longer time horizon means more uncertainty, this difference of strike price + premium - stock price is known as the time value. Since this works out to zero in your hypothetical, people are saying that there is no time value, and remarking on how unrealistic this is.
Getting back to your original question, you're right. Just as there's arbitrage between either of the options and the stock, there's arbitrage in buying A and writing (selling) B. There's no situation where B is better than A. Since both have the same value for strike price + premium, that means that if the stock goes up, it doesn't actually matter which one you have. But as long as there is a chance of the stock going down, A is better than B. If the stock craters, then you'll want to let the option expire unexercised, losing just the premium. Since A has a smaller premium, its possible downside is smaller.
You also said that you've seen both options being offered, and are wondering why anyone would take B when A is available. That is indeed odd, and calls into question whether there's some further information that's missing.