AITAH for "tracking" my girlfriend

A couple of months ago my (50) now ex girlfriend(34) (This is not why we broke up.) got into an accident in her car. happened to have just left her so it was easy for me to come back and get her. She'd called her insurance and a tow truck but had a problem. She said she couldn't cover the deposit for a rental. So I offered to let her borrow my car until hers was repaired, since I rarely use it, anticipating the repairs would take several weeks to complete.

One day I was looking for my keychain because I needed my door key, and couldn't find it. (My memory sucks so this isn't an uncommon occurrence.) Without thinking, I opened the Tile app on my phone to use the find my keys function, which plays a sound on the fob on your keychain. It just didn't click that I didn't have the keychain with the Tile on it but had the house key in the change tray..

When I opened the app and clicked on that tile it showed my keys were not in my house, which obviously makes sense. What's didn't make sense was that they were at a football stadium. Since she was supposed to be at work, (She'd literally just texted that she was on her way to work.), and we'd discussed that we couldn't go to the game because because she couldn't afford to take off, I was curious what was going on and texted her, explaining the situation and that I wasn't happy about being lied to.

She went ballistic, saying I had no right to "track" her, and that I should have told her about the Tile. I said it's right there on the keys so it wasn't a secret, and it didn't even occur to me to say anything because I don't even think about it being there until I can't find my keys. I also said nobody was "tracking" her. I hadn't even opened the app until right then. I also pointed out that it is not like I cooked up some nefarious plan to "track her" in the ten minutes it took for me to get to her the day she wrecked her car. The Tile was obviously already on the keys.

She said it was a huge violation of her privacy and I should have told her about the Tile. I pointed out that even if you have an issue with the keychain, the car, like many other cars, has a GPS tracker in it, so I could have "tracked" her without the keychain. It shouldn't be an issue for me to know where my own car is. I also again pointed out that nobody "tracked" anyone. I literally found out just because I couldn't find my house key and opened the app.

So, Reddit, am I the asshole for not telling my then girlfriend that the black thing on my keychain, which she had because I was letting her use my car for a couple of weeks, makes it possible to know where the keys are if I lose them?