I was wondering if it's a good option to buy a gift for someone else, or are the fees too high and it's not worth it?
It all depends what your other options are. One of the biggest advantage of BTC ATM is that they often don't ask for any sort of documents, but then again that depends on the country you are living (in my country I can do it completely anonymously). Regarding fees, they are generally high so if anonymity is not your priority, there are much better ways to buy and sell Bitcoin.
Also I'm not sure how it works but figuring they give you a private key, and how safe is that?
No, you don't have too give your private key, and anyone asking you for that is scammer. The way it works is simple: you choose on ATM the amount of BTC you wanna buy ( for example $100 worth of BTC), then the machine scans qr code of your wallet so it has the address where BTC will go, and in the end you insert the bills in the machine and that's about it. BTC will be sent to your address and you will get a receipt.
About my personal experience, I've been using BTC ATM exclusively for several years, but when fees became too high for me 9iirc 2 years ago) I decided to find other ways.