Air Carrier Captain reported on a visual approach; responding to an EGPWS alert on an antenna that was clearly not a factor resulted in crew executing Controlled Flight Into Terrain (CFIT) maneuver.

Narrative

Flight from ZZZ to ELP. Captain was Pilot flying (PF). While on the ground in ZZZ; received ATIS for ELP. Due to winds at 080/12; visual approaches to Runway 8R were in progress. Crew loaded and briefed the visual approach. Clearance was MNZNO3. We briefed on the ground prior to departure; the visual approach to Runway 8R into ELP; due to short flight. Routing was perfect for the briefed approach because of over flight of the mountains from the West to ELP VOR then to CHUCO intersection for the visual routing to 8R. While en route; ATC re-cleared us for the WAZKO arrival; which brings you over the Rio Grande; and just southwest of the mountains valley separating El Paso from Mexico. Received radar vectors prior to KOACH intersection; then cleared for the visual to Runway 8R. PF selected HDG SEL; and maneuvered around the northern mountains to set up for a dog leg from the South to a right base to Runway 8R. While maneuvering; received an aural Enhanced Ground Proximity Warning System (EGPWS) caution obstacle. Both the PF and PM saw the tower to the left approximately 1 mile North. We immediately and unequivocally determined obstacle clearance was not a factor; and we both stated so. Shortly after; we received EGPWS warning obstacle; obstacle; pull-up; executed maneuver; pilot response to EGPWS alerts. At no time was the crew or aircraft in danger. However; the safest course of action was to execute the maneuver. Aircraft was fully configured for landing and landing completed; PF pushed TOGA levers. As soon as the aircraft pitched up; the warning ceased. ATC allowed a return for landing Runway 26L with a 10 kt. Tailwind. Recommend to note the difficulty of the WAZKO arrival while conducting visual approaches to Runway 8R. I will not accept the WAZKO arrival in the future.