Hi CreativeRaul,
The Cylinder is based on a “generated” object, a “Primitive”. More often than not, these objects are sometimes mathematical patchworks.
To overcome this, you need to make them editable, [C], and if they end up in several Polygon Objects, the Object Manager>Objects>Connect Objects (and delete) is needed.
Then use the Mesh>Commands>Optimize. Check out the Command by selecting the Cogwheel from the menu. Note: everything that is selected during optimizing will be optimized, and what is not selected will not, if points or edges or polygons are selected. You need of course select the Polygon in the Object Manager at any time to be able to “optimize it”.
Optimize means here, that the points of a mesh, with the same position (tolerance!) will be merged to one. After that, the Bevel Tool will work more like you want it to.
All the best
Sassi