Marko, the Stockcharts info is misleading you. Typical price is not smoothed. In fact the way it works (doubling the effective move at bar extremes) is almost the opposite of smoothing.
Most of it (SMA and normalizing factor) is smoothed but that is just the
background position vs SMA and normalization to the 200 to -200 range. It's smoothing is a "good" thing because it stops the average value (average price) and the current normalization from interfering with the output calculation - they become very near to constants for the calculation.
But the key output, Typical Price (H+L+C) isn't smoothed. It has a near constant subtracted from it (the current sma) and is divided by another near constant (the normalizing factor) but it isn't significantly smoothed by them.
If it was the ma of TP then it would be smoothed (as happens with most indicators). But as it is the impact is that if the close is not at the extreme then if it moves X% it moves CCI by X% over 3 and if its at the extreme it moves CCI by 2*X%/3. So at bar extremes CCI pushes twice as hard as it does "within" the bar. There is some small distortion of this because the normalization and average will "catch up" towards it but that effect is very small.
The lack of smoothing of TP and the multiplying factor at bar extremes is why CCI is jerky while stochs etc with the output price component smoothed are not. That's also why you see people putting up smoothed CCIs - they prefer the smoothed (summarized) version.
Note: despite debating the construction of an indicator I stand by my original recommendation for new traders
in this post. FWIW ... anyone who wants to use an indicator should do the work required to really, deeply, understand how they work. The work will probably put you back to PASR with horizontal lines only.