Being alone in the wilderness and being there with a group or with friends are two different experiences, each with their own merits and drawbacks. You have experiences with groups already, so I'll talk about going out alone. I do this much more often than going into the wilderness with others. When I have only a little time to go into the wilderness, I'd rather be by myself.
- You feel much more aware of your surroundings. Other people talk, distract, make noise, etc. All that makes it harder to really perceive the wilderness, and it will also alter the wilderness itself. You're far more likely to get to see a bear in the wild, for example, when alone than in a group.
I have only encountered a bear (talking North American black bears here) while hiking three times, and I was alone each time. I'm sure I've gotten close to bears many more times than that, but these three times I noticed the bear before it noticed me. Each time the bear bolted as soon at it saw me. Getting close to a bear before it notices you with a group is basically impossible.
Groups as a whole, and often unfortunately individuals in groups, just aren't careful and quiet, and don't care.
- You can do what you want. With a group, each person is sortof obligated to not cause the group to stray too much from its stated objective. That means if you come accross a scene you'd like to take the right photograph of, you can at best take a snapshot or two. Carefully moving around to get the best vantage point, trying different approaches, etc, that take time are out of the question. Good nature photography takes patience, which is something you can't have with a group. Generally I don't even bother take a camera with me on a group outing. For me anyway, nature photography is a personal experience between the wilderness, myself, and my camera. Having other people around seriously interferes with that.
- You can go the pace you want, whether fast or slow. Groups tend to go slower than individuals. That can be frustrating if you just want to hike. On the other hand, it keeps you from stopping and exploring something in more detail if you want. You can have some interesting experiences just sitting quietly somewhere for a while.
Try it some time. Go somewhere people aren't regularly coming by and find a comfortable spot, like on a nice rock or against a tree, with a good view. I like overlooking a wetland. After just a few minutes you'll be surprised at all the comings and goings of the wildlife. You can't do this in a group because there is always someone that can't sit still and shut up for 30 minutes, and each person is one more chance people will be spotted and the wildlife will be more inhibited.
You don't need the deep wilderness for this. Some of the best experiences I have had like this have been right in the town I live (Groton Massachusetts). We have about 10 square miles of wild land open to the public, but it's a patchwork of many small parcels. Almost all of it is within 1/2 mile of a paved road. However, there are still some very pretty places where people go infrequently enough that you can sit and watch nature (actually more like experience nature) by yourself for a hour at a time. You should definitely try something like that. When you've had enough, you can be back in the car in 10-20 minutes.
- You don't have to have a plan. I occasionally lead hikes for the AMC and for the town Trails Committee. These are all advertised as going someplace specific, at least roughly. People aren't going to show up to a "We're just going to bump around, don't know where we're going or when we'll get back" hike. When going out by myself, particularly locally, I start someplace and then just go whatever way I feel like going when a choice comes up. That may be related to the weather, where I haven't been recently, something looks interesting in that direction, I see something I want to check out more closely, or just I feel like it for no specific reason.
You mention loneliness, but that is completely foreign to my solo wilderness experience. I'm way too busy and engaged to be lonely. Besides, lonely is a long term thing, not something you experience over a one day or even a few day outing.
It seems your real issue is being scared of being out there alone. There is a small element of rational fear there, since there are dangers and the consequences can be much worse without other around to help. However, these are likely blown way out of proportion in your mind. If you have a true phobia of being alone in the wilderness, then doing that is probably not for you. However, if you are capable of addressing it rationally then there should be no problem.
Actually think about it. What is that sound you hear outside your tent at night? What does it really mean? What is the worst case scenario? When you think about that, you'll realize that anything not astronomically improbable is no big deal. At night you will hear things outside the tent. You can eliminate anything up in the trees as a problem right away. Most of the rest is merely rodents poking around. Try the reverse. See if you can spot whatever it is. Betcha you can't. By the time you get up, open the tent, and shine a flashlight, whatever made the noise will be hiding.
If you hear something heavier, then it gets more interesting. Big things are less likely to be able to hide, and if they run away you can hear that. Sometimes you can get a glimpse of a bigger animal at night. One time I could hear something definitely bigger than a mouse making its way closer and closer to my tent. It sounded like it was checking out nooks and crannies as it was rustling a lot and moving slowly. I was guessing raccoon, but when it got close enough that I thought there might be a chance to see it before it ran off, I opened the tent, shined the flashlight, and found a porcupine. Porcupines have evolved such a great defense that they aren't much bothered by other animals being nearby. It basically continued bumping around like it had been, sticking its nose in the leaf litter here and there. It actually seemed to be using the light from my flashlight to its advantage!
I have heard what I was pretty sure was a bear outside my tent a few times, but as much as I would have liked to have caught a glimpse, I thought it better to lay still for a few minutes until its foraging made it go past. With the right mindset, these things are fun and interesting, not scary.
I do admit to being irrationally spooked once. I was in the Kofa National Wildlife refuge in Arizona and driving on a obscure dirt road when I spotted a cougar (Puma concolor), also known as puma, mountain lion, and about a dozen other names. That wasn't what spooked me. In fact I thought it was way cool. I tried to see where it went after it bolted upon seeing the car. I felt lucky to have seen a cougar in the wild. However, the next few nights when getting into my tent I was spooked. Now I know the chance of a cougar being around are exceedingly slim, and the chance of it not running away if one was around even slimmer. But, it still took a few nights to get over not being spooked whenever getting into or out of my tent in the dark.
Basically I think the lesson is that the more you feel comfortable that you know what you are doing, the less you will feel afraid of what might be out there. The only solution to that is probably going out and doing it, and then things will feel more comfortable because they are no longer unknown.