The most significant and ongoing expense drinking coffee is the coffee. If you make it at home you’re going to beat the coffee shop price every time. No argument. Whether you hot brew or cold brew, you still have to purchase coffee. Sure, you can cold brew in a Mason jar, but even a cheap drip hot brew machine will last years and so amortize the cost to pennies or less per day. But you’re still going to buy coffee. If you’re content with supermarket house brands, you’ll pay the least. But if you really like coffee and want to experiment with different brands, different grinds, different roasts, different blends, etc, cold brewing won’t be any cheaper that hot brewing. And with cold brewing simply because you can tell the difference between all those different coffees on offer, you will eventually want to try them.
As for overall convenience, cold brew certainly has the potential to be convenient but lots of folks would think it hard to beat adding water and dumping a scoop or two of coffee grounds into a Mr Coffee machine.