Kind of ironic that video has an ad for a "mental hygiene" app... when a good first step would be not watching that sort of videos... but let's go point by point:

• Logistics: that's SpaceX. Starship is planned to carry up to 100 people at once. As for trip duration, think XV century ships on multi-month long trips: tough, but survivable. Will need unmanned supply trips, but as it says, some could fail, just send more. Manned trips... if they're full of volunteers, some could fail too (plenty more where those came from).
• Justification: because. Same as always: adventure, fame, science. Probably no raping indigenous population and robbing them of gold, but still. People get into F1 cars, skydive, free climb, etc. Some die, that doesn't stop the next ones.
• Living on Mars: yes, probably in underground habitats. Definitely not "just the 1%", the plan is to have indentured labor, so everyone has a chance! (to be a slave, but on Mars) Premature death... see previous point.
• Psychological effects: with a few hundred people, same as in any small society. Not everyone needs to be surrounded by billions of people 24/7.
• ~~Terraforming~~: red herring. Musk doesn't have plans for that, so less BS.
• Alternatives: night on Antarctica is 4320 hours, on the Moon it's 336 hours, night on Mars is 12 hours and 18 minutes. See the difference?
• ~~Mars One~~: red herring. Not a Musk project, scammy from the beginning. That's two totally BS points.

It's a risky plan, but still a plan.