Videos? Yeah, for sure, if you saw it on TV or on the Internet it must be true ......
More seriously, if you saw a video of racing, those guys don't carry tools, they carry trains of cars carrying tools, wheels, mechanics, and spare bikes. If you saw a "How to" or "Look a this ride" video, the rider had a film crew, with cars, spares, etc .... otherwise a flat could have ruined the shoot.
Pretty much every time I get a bike off the rack I bounce it a little to see what rattles, and press the tubes with a thumb or finger. If I have ridden it the day before, that it all, as a rule. If it has been on the rack for some time I do a safety routine---squeeze the brakes, bounce front, rear, and whole bike, shake the bars, shake the cranks, maybe wobble the wheels, and squeeze the tires. I don't tend to lose enough air overnight to have to add any every day ... not sure I could tell 93 psi from 95 psi.
Since I don't have a support staff, I carry tubes, tools, food, pump, anything which might help me get home in about 95% of situations. I have had to walk home a few times, but surprisingly few .... and since I don't have a go-to "Call of shame" recipient I definitely would rather fix it roadside.