Also, to add to the other poster's point, in a medieval siege, the defenders have every reason to believe the attackers will happily let every man, woman, and child behind the walls die gruesome deaths to starvatiom or disease. That's why, when it came down to the wire, cities would submit.

In modern times, cultivating a believable military posture of, "Surrender, or we will personally execute every last motherfucking one of you" is politically dicey. Look at the news stories coming out of Gaza about supplies running low thanks to Israeli interference. Right, wrong , or indifferent, the international community (as well as your domestic community, if those that disagree with these sorts of tactics are allowed to make their voices heard) tends to look down their noses at targeting noncombatants populations. So, due to these complications (which were largely absent or less impactful from warfare in the time of Genghis Khan) wholesale slaughter of civilian life isn't really openly used. In fact, guidelines like "proportionality" are invented which dictate the level of response you can give certain provocations and what not.

So, if you're a modern day commander being tasked with taking an urban center, the closest way to approximate a medieval siege would be to absolutely carpet bomb everything. Make it known that you will happily let every single person in Moscow die, if not send them to the afterlife yourself. While you're bombing the suburbs, you'll also need to encirce the whole city to prevent supplies from being delivered, since you can't guarantee every bomb will hit it's target and need starvation to provide additional assurance to the population that, if they maintain their current course, they are doomed.

Unfortunately, the world isn't going to allow that, and you know it, so you commit to the level of bombing deemed acceptable by the world at large. At best, you wind up in a situation like London during the Blitz. Your bombing runs are effective, in that they disrupt the daily life of citizenry, and cause some infrastructure damage and loss of life. However, you're never going to be allowed to scale up to the point where your victims feel they have no way out but to submit. There's enough plausible deniability that, even when a bomb hits close to home (literally or figuratively), the victim is more pissed at the bomber than their government.