But that applies to your 1st few games only? After 10 games, it doesn't matter anyway?
Yes, but the first 10 games matter in themselves for one reason: the first impression of the site. If a new casual player loses many games in a row to start out, they may think the site is too difficult and leave, and obviously the site wants to avoid that. Starting too high will also make their "max rating" statistic be higher than anything they could actually obtain.
And in the long run, if there are still people like you, then why wouldn't they be harmed from choosing 1500 given that '1500' will actually be 400 points stronger than the previous '1500'?
Lowering the initial rating will, over sufficient time, cause the established players' ratings to be lower than if you hadn't lowered the initial rating. That does not, however, necessarily mean that the established players' ratings will be any lower than they used to be. There are various inflationary and deflationary factors; initial rating is just one of them.
I mean in 2023, who has a FIDE rating but not an account in both chesscom and lichess?
This got me to thinking. When you create a chess site you initially get players of all strengths. But if all the strong players already have accounts because you've been a popular site for many years, wouldn't that imply that the average player making a new account now is weaker than the average player making a new account when you started the site? Especially if, say, a pandemic caused people to have a bunch of free time and a miniseries increased the popularity of chess among people who hadn't played much before? It's entirely possible that, because you're attracting a different class of player, the initial rating might have to be adjusted to keep the current players at the same level. (I do not know if this is indeed the case; only that it is possible.)
But one advantage of letting the players select is that variations in strength are somewhat accounted for. If the NYC public schools suddenly put chess in the curriculum and forced every one of their 920,000 students to make an account, the kids could set their rating to 400 or 800 and that would reduce the impact compared to if they all started at 1200.