How to add music to my iPhone 16 from my PC without deleting existing songs?

If your iTunes deleted the music that was on the iPhone, that suggests that the original music was added to the phone by a different iTunes Library. The "different Library" scenario can occur when using a different computer or if iTunes has been reinstalled (not simply updated) on the same computer, or if the computer has been repaired.

iTunes is designed to synchronise with one library. If you attempt to sync it with a different library, iTunes puts up a warning to tell you that if you proceed with that sync it will remove the existing content. I assume therefore, that this is what happened.

I'm not too familiar with the Apple Music service, but as far as I know, if you're using the Apple Music subscription service, it tends to override your library (your purchased music).

When one uses Apple Music in iTunes on a desktop your own purchased library is seemingly, not available to you. However, I found a video on YouTube which suggests that on an iPhone, your library is available, although how one gets it onto the phone from iTunes I don't know. (I still use purchased music, I do not use the subscription services.)

Once on your iPhone, go into the Music app and look for the Library icon on the bottom bar:

The screenshot above shows the Library icon at the bottom. It's in red because that is currently the selected option, which is shown above it. In theory, all the music that you own and is on your phone can be found in these headings.

See whether this Apple article helps: Redownload music from the iTunes Store on your iPhone or iPad - Apple Support. It covers music purchased from the iTunes Store, but not music purchased elsewhere (such as Amazon, Bandcamp or from CDs). I'm still puzzling over that and I'l, let you know if I work it out.

Also consider having a look at Apple Music on Windows - Apple Community where others may be able to offer a better solution than I can.