When you're talking about safety, hotels should do everything possible to reduce the risk of a guest getting robbed, attacked, or raped. Having hotel staff verbally announcing room numbers does increase that risk, whether you want to accept that as fact or not.

For the record, I used to work in hotel, including at the Front Desk. While robberies and attacks are rare, angry spouses calling and/or showing up and demanding to know if their partner checked in happens more often than people realize. That sort of situation happens when people cheat on their spouse or when there are instances of domestic violence.

Aside from safety, announcing room numbers increased the chance of financial fraud. Many hotels allow guests to charge various things to the room solely by writing down the name and room number. (The OP seems to be describing that process.) A scammer could hang out in the front desk area where hotel staff are announcing guest room numbers and surreptitiously get the name off of luggage. They then turn around, charge something, and leave.

Last year I arrived early at a Disney World area hotel. While I waited for a room to get cleaned, I was working in the lobby next to the concierge desk. I watched the woman sell dozens of theme park tickets. Everyone charged them to the room. It appeared the only verification required looking up in the computer to see if the name and room number matched.

A scammer could have easily charged a bunch of one-day tickets and gone immediately the theme park to use them (or scalp them). While all illegal, that's not the sort of thing that would get investigated by the police if it's discovered and reported by the hotel. Police don't generally have time to deal with every case of petty theft.

TL;DR: No hotel employee should ever announce a room number.