Intuitively, why doesn't always picking unpopular integers lower Probability of winning lotteries?

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What’s wrong with my reasoning in purple? Please pinpoint which sentence and step fails. I seek intuition. I DON’T want proofs, or formal arguments. Thanks!

Presuppose lotteries let players pick any integer N≥32. Picking unpopular integers lowers your probability of winning lotteries. Why? Winning numbers range randomly from 1 to N. But you artificially narrow yourself to [32,N]. By disregarding [1,31], you flout the lottery’s random distribution of winning integers! As [1,N] contains more integers than [32,N], picking numbers ∈ [1,N] proffers more chances to win than picking ∈ [32,N]. Q.E.D.

Here’s something you can do that might have an impact on your game: studies show that most people play numbers based on special days of the month, such as birthdays and anniversaries. So, numbers greater than the number of days in the month, or over 31, are less frequently played. If you do win with a less popular number, there may be less people to split a prize. So while picking these numbers won’t increase the odds of a win, it might increase the amount of a win.

Anyhow, if you do play the lottery, you should _not_ play obvious combinations, or anything that can be described in one sentence, such as "the birthdays of my pets, increased by 5" (yes, there is going to be someone who has pets born on the same days as you, and who also thinks 5 is his lucky number).

The only thing you can do to improve your situation is to pick an unpopular set of numbers to reduce the number of ways you split the prize should you win.

You should also not pick the date - or any date for that matter.

The arguments of picking unpopular numbers only matter if there is a jackpot that is divided among the winners. In that case you want unpopular numbers so you share less.