Why does the army have regiments in their unit designations but no Regimental Commanders?

The Army was previously arranged that way, similar to how the Marines still are today. The issue was that when it came time to deploy, these regiments had to pull their support personnel from other units. So the Army came up with a more modular system of the Brigade Combat Team in order to have an organic unit that is readily deployable. They pulled all those support personnel into the BDE and intermixed them into the battalions. They broke up the regiments and mixed different combat arms together to create a combined arms combat team. Now, instead of pulling a few battalions of infantry, some scouts, engineers, artillery, FOs, medics, etc they are all organic to the Brigade and always training together.

The senior Battalion Commander in the Regiment, serves as the Regimental Commander. I believe the 1-506 IN BC is the 506th Commander. Their CSM is the regimental CSM.

The 11Th, 2nd and 3Rd Cavalry Regiments fight as regiments and have a Commander

The Army with minor exception does not fight along regimental organization lines. Even the 75th doesn't. As they deploy Battalions and Companies as part of a CJSOTF or a JSOTF.

Edit: in my haste, I forgot to mention that if the BDE lineage is with that regiment, then the BDe CDR is the Regimental Commander...example 4BCT 101st was aligned with 506th IN, so Currahee 6 was the BDE CDR. When they deactivated the BDE, the Regimental command passed to 1-506. 1-506 and 2-506 passed to other Brigades in the 101st, 1st (327 IN ReGt) and 3rd (187 IN Regt) while retaining the 506th lineage. This is specifically tied to lineage and the US Army Regimental System.

The reconstruction of regiments is not possible under the current rules. Different units are activated and deactivated based on an TIOH Order of Merit List. So different regiments/battalions are pulled from the next one, not the one that goes to that Regiment/BCT based on unit type required. As a consequence, you had a BSB that was lineage wise the Division Ordnance Company then eventually the MSB, redesigmated in place. 1-61 CAV as the RSTA Squadron, which was a Tank Destroyer Regiment, but he infantry battalions were HHC, 1-506 and 2-506 all 506th. Then MG Petraeus got an ETP to do that, becausemof the history. The 506th brought out of retirement ahead of others.

We do, 2nd BDE/82nd ABN Dv, is also the 325th Airborne Infantry Regiment Commander- the oldest unit in the 82nd.