I definitely feel better overall when I ride my bike regularly. My mood is better, I have more energy, and I sleep better. I got into it around the time I separated from my ex-wife, and I think it's a significant part of what kept me from falling into a serious depression when I got divorced.
I hate running. I hate swimming. I hate lifting weights, or anything in a gym. Hiking is nice, with the right company. But I enjoy biking around my neighborhood and along the city greenways. I was wanting to be more active, healthier, etc. but couldn't find anything I actually would stick with until I found an activity I actually enjoyed. Lately I've been feeling pretty down, getting uncomfortably close to a major depressive episode, but riding my bike for a while always makes me feel a bit better, at least while I'm doing it. I think not biking, due to it being winter, is indirectly part of why I've been down - I just haven't been doing much that is enjoyable while dealing with a lot of stress including some major life changes.
Fwiw, I'm a psychiatric nurse. I've read the science news articles that say exercise is better than anything and the ones that say it doesn't do much. I've also read some of the actual research/journal articles. The reality is that we're not really certain how much difference it actually makes, but it probably depends on a variety of factors that vary from person to person and based on the type of exercise. We do know that people who are physically healthier also tend to feel better mentally. What I have seen personally, and experienced myself, is that exercise helps with mood, but it's not a cure-all, and it's not instant, and it makes a difference if it's something you enjoy doing independently of it's health benefits. Exercising outdoors, particularly in green spaces, probably also helps.
You should try biking in the winter. Get a second bike, like a rigid alu MTB with platform pedals and studded tires, layer some winter clothing, and go to town on salty slush, fresh snow and ice. It can be exhilarating!
Sounds like it could be interesting to try snow biking. Doesn't often snow here, and when it does it rarely sticks. Maybe 1-3 actual accumulations per year, so my gravel bike is suitable practically all the time. It's just the cold that makes it unpleasant to me - layering when it's 50F is fine, in the 40s F is doable if it's sunny. Anything below that is just miserably uncomfortable to me because of cold wind to the face and the need to adjust layers as I warm up and cool down.