Is Mountaineering a Profession or a Hobby?
It can be either. If you get paid for it, then it’s a profession.
If you want to make it a profession, consider the various aspects of it that are potential revenue streams, such as: guiding others, assisting others with planning, getting sufficient reputation for yourself to get sponsorship(s), writing books/travel guides/planning guides/blogs/video journal channel/…, use online presence to get paid for referrals to outdoor supply vendors, connect with local chambers of commerce in the communities where you plan to be and pursue opportunities to get paid to help their tourism business by promoting their local area on your blogs/video journals, whatever.
I’m sure you know more about mountaineering than I do. The trick is to take a step back and consider the sport from a broad business perspective. Then pick an area you feel comfortable tackling as a means to get revenue, then go on to the next and the next until you have developed a business you’re comfortable with. If pursuing avenues to generate revenue is not something you’re interested in, then you can consider being an employee of a company that provides mountaineering guides. If neither of these avenues is to your liking, then enjoy your hobby.