There are several methods to propagate hydrangeas: Division (through suckers), Cuttings, and Layering.

Division through cutting off suckers is a viable propagation method. The drawback of this method is once the plant is cut from the parent plant’s mature root system, it may go into shock. Carefully move the dirt away from the sucker and uncover back to the parent plant as close as you possible can. This sucker should reveal some roots. Clip off the sucker at the base of the parent plant, then re-plant in your landscape or in a container of soil. Watering is the most important thing. Don’t let the new roots dry out, but keep the soil moist but not soaked. Again, the sucker trying to grow on its own, immature roots will likely shock the plant. Although this website is from England, the information is valid when it comes to suckering-type plants:
https://www.rhs.org.uk/propagation/suckers

You mentioned you had read about the Cutting method. Hydrangeas are successfully propagated by softwood cuttings. https://extension.missouri.edu/media/wysiwyg/Extensiondata/Pub/pdf/agguides/hort/g06970.pdf

The final method of propagation is the Layering method. This requires selecting a branch close to the ground and removing any leaves that touch the ground. Scrape off a small section of bark on the underside of the branch, and layer it in the ground and cover with soil. Some people put a stone or brick on top of the soil to keep the branch securely in the ground. Roots will form and the branch will become a new plant. Cut it away from the parent plant, but leave it in the ground until it because a hearty and sturdy plant on its own, then dig it up. (see photo attached)

I hope this information is helpful.