Is this little charred cylinder near my TV power input a fuse?

That FS1 component is a 3.15-amp, 250-volt **slow-blow fuse** (the "T" stands for time-lag). The loud pop and heavy soot mean it failed violently, which almost never happens on its own. Fuses don't commit suicide; they're murdered by a short circuit downstream.

When a fuse detonates like that, you have an aggressive dead short on the primary side of the power supply. Nine times out of ten, the main switching MOSFET or the bridge rectifier diodes shorted out completely, dumping mains AC straight to ground until the fuse blew. If you just solder a new fuse into that spot and plug the TV back in, the new fuse will vaporize in a fraction of a second with another loud pop. On top of that, that black soot on the PCB is conductive carbon, which creates its own shorting path across traces until it's thoroughly scrubbed off with isopropyl alcohol. Component-level troubleshooting on these boards usually isn't worth the hassle unless you allready have a soldering iron, a multimeter with a diode check mode, and enjoy hunting down shorted silicon. Even if you swap out the blown transistor, whatever gate resistor or driver IC pushed it over the edge might still be dead.

Your best move is to look for the part number silkscreened directly onto the board—often something starting with "BN44" for Samsung, "EAX" for LG, or a standard Vestel code—and search for a replacement board on eBay. Swapping the entire power supply board usually costs twenty to forty bucks and takes about five minutes with a screwdriver.

Before you order a replacement board, have you ruled out an external surge coming trough the console?

Since your kid was actively gaming when it popped, I'd wonder if a spike traveled right through the HDMI ground. If that's what triggered it, the main logic board might be fried too, and a new power board won't bring it back. Did the gaming console survive, or have you checked if any other gear on that strip took a hit?

Back in my CRT days, whenever a chassis blew a fuse like that, we’d hook up an old 100-watt incandescent bulb wired in series with the AC cord before flicking the switch. If the bulb glows full blast and stays lit, you've still got a dead short staring you down. If it flashes once and goes dim, your circuit charged up normal.

Saved me a mountain of blown fuses over forty years on the bench. Just watch yourself around that big main filter capacitor. Modern switchers store a nasty kick, and unlike the old linear sets with heavy iron transformers, that charge can sit there waiting to bite you. Treat it like a loaded gun.