If we all stopped using fossil fuels, oil execs would invest in something else and continue to be wealthy.
Exactly! They would abandon control of a product which is no longer valued and instead control a different product.
Remember that the wealth of your typical executive is mostly tied up in assets. They don't have giant vaults of money in their building like Scrooge McDuck. They own shares of a company, the value of which is determined by the market. The newspaper reports you see giving dollar values to their names are just converting the company share value. Mostly these assets are diversified into several areas, some higher and lower risk.
Say I was a CEO. My primary wealth is the business I control, of which I own 25% and the rest is split up with other executives and shareholders. This business represents 50% of my total wealth, from which I pay myself a large sum approved by the other shareholders in the company. I also own property and 'safe' stock in cash and gold, etc. which represents the other 50% of my wealth.
Suddenly someone opens a business that makes my product worthless. The value of my business drops to nothing overnight. 50% of my wealth is now gone. Realistically this never happens, but if I think it would happen I could sell my share of the business and buy something else instead that wouldn't devalue.
The fastest way to grow wealth is to use existing wealth to create more. Investing is a common way to do this. I give you my money now to use to make your business better (hire staff, buy raw materials etc) and in return I take a share of the profits proportional to my investment. If you want the profits to be yours again you buy back my share. This is a stupidly simplified example but it gives you the general idea of my point. The only way that my wealth or yours grows, however, is when the market demands the product we're producing.