Rooted Android device and more regular updates? My current Android phone Samsung Galaxy S4 (GT-I9506, Android 5.0) has much outdated software and OS. I would like to get much faster update cycle. The current Android and Samsung support cannot provide it. Therefore, I am thinking about rooting my device and possibly this way speeding up the update cycle. 

I have installed CyanogenMod in my unrooted device unsuccessfully a few times. I think the failure may be due to the unrooted device. I think you cannot get the full control of the device without the rooting and thus insufficient updates in your device.

Can a rooted Android device have more regular update cycle? However, I am not sure if rooting can make the update cycle worser because of possible loss of the main developers' support in the device and OS.
Generally, if you're talking about updates for any ROM, then you should know that it doesn't matter if you're rooted or not to get regular updates.

Moreover, rooting modifies your /system partition, due to which OTAs (for stock ROM) will fail to install, because nowadays most updates are Block-based OTAs.
So, for successfully installing official OTAs, it's better not to root your phone or use systemless root.

You're not getting regular updates because maybe Samsung hasn't yet released any updates for your device.

But if your phone is rooted and if you have flashed a custom recovery, then, as you said, you can install any custom ROMs with active developers which might be getting regular updates.
As @Tamoghna said, you can install ROMs like CyanogenMod which might provide daily updates.
You can also check for CM daily updates using apps like CyanDelta Updater (as said by @Firelord) which provides delta updates of minimal size.