That sound is definitely the starter clutch letting go. The starter clutch consists of a set of spring loaded rollers in the flywheel that grab a boss on a gear that the starter motor turns. When the starter gear is turning faster than the flywheel the rollers grab and it turns the engine and when the engine starts and the flywheel turns faster than the starter gear the rollers disengage.
Your starter clutch might be developing problems but it sounds more like the starter motor isn't turning fast enough to me. That might mean that the starter motor needs to be cleaned but I would check the condition of the battery first (it might just need to be charged).
Welcome to the forum and welcome to the world of antique vehicle ownership (they own us, not the other way around). Your bike is about 4 decades old and may or may not have had all of the maintenance necessary to keep it safe & reliable so it is highly recommended to download the Factory Shop Manual for your model (available through the CX Wiki - link in my signature) and go through all of the service procedures, regardless of whether your bike has reached the specified mileage.
I also recommend looking on all rubber parts with suspicion because rubber does not age gracefully. Check the date codes on your tires and replace them if they are over 5 years old no matter how good they look & feel (old rubber simply cannot flow around the irregularities in the asphalt well enough to grip, especially if it is cool or wet). If your bike still has the original rubber brake line(s) (should be replaced every 2 or 3 fluid changes = 5 or 6 years) I recommend shopping for modern stainless braided ones (they last practically forever and double the life of the fluid). And don't forget things like the rad hoses and the boot between the engine and swingarm (they can crack on the bottom where you don't see it).