That clicking every few minutes with a freezer that won't hold temp is a classic sign the compressor is trying to start and failing. The click you're hearing is the start relay kicking in, then the overload cutting it back off when it doesn't get going. It'll do that over and over. Could also be a bad relay on a healthy compressor, which is actually the cheaper outcome, but the pattern you're describing usually means the compressor itself is struggling. Either way it's squarely in sealed system / compressor territory, so yes, the phone rep's guess was reasonable.
I went through almost the same sequence with mine a few years back. First visit was diagnostics only, tech confirmed a dead compressor, ordered the part, second visit did the install. Two visits total and it was fixed. The three-step thing only kicks in if the first repair doesn't solve it, so don't assume you're signing up for a month of appointments. Ask the tech on visit one to log it explicitly as a sealed system issue if the compressor is the suspect. That matters because LG treats sealed system claims differently from regular parts, and if it eventually needs to escalate to a replacement, having it classified correctly from day one saves you from redoing arguments later. A few things that helped me: get the case number and the tech's written diagnosis, keep every appointment confirmation in email, and take a photo of the model and serial plate before anyone shows up. If the first repair doesn't fix it, don't let them start over from scratch with a new diagnosis, push to have the original case reopened. Meanwhile, get a cheap appliance thermometer for the fridge side and keep it under 40 degrees if you can. If it's drifting warmer, move the dairy and anything else perishable to a cooler with ice packs until the fix happens. Milk spoiling early is exactly what you'd expect from a fridge sitting in the high 40s. As for wheather the multi-visit process is an LG thing, it's mostly just how manufacturer warranty service works in general. They use third-party shops, and the authorized techs can't order sealed system parts without a documented diagnosis. Independent repair guys can sometimes skip the whole dance, but if it's a compressor under the sealed system warranty, use the warranty. That's exactly what it's for.