Concern About Date Change on My New Yacht-Master RLX Titanium (226627)

[Tit4life]

Hello everyone,
This is my first post here, and I’m hoping to get some guidance.

I recently bought a Yacht-Master RLX Titanium (226627) from a grey dealer (not an AD). (Yes, I paid a hefty premium—but honestly, I love the watch, and it checks all my boxes. This is my first Rolex and probably my last—it's the one for life for me!)

After getting the watch, I started watching some videos about its movement. That’s when I noticed something odd: when I manually advance the time, the date doesn’t consistently change at exactly 00:00. Sometimes it switches at around 23:59 instead.

This got me a bit worried. From what I’ve heard, Rolex movements are usually spot-on with these things, so now I’m questioning whether this is normal or if there’s something wrong with the watch. Since I purchased it through Chrono24 from a shop seller (it came as a full set and brand new and with some stickers on it), I’m also wondering if I should be concerned about its authenticity.

Here are my questions, separated for clarity:

Is it normal for the date to occasionally change slightly before 00:00 when adjusting manually?

Should I consider sending it for a check or service, or am I overthinking this? (The watch is just a few months old according to the card.)

For these two, I’d love to hear your input regardless of the following concerns.

Is this something that could indicate the watch is fake?

What’s the best way to verify its authenticity?

I might be overreacting, but I’d love to hear from more experienced collectors here. Any advice or insights would be greatly appreciated!

Thanks in advance!