Okay, I understand the situation. Let's break down what you're describing and whether it meets code.

What you're describing

You're describing two variations of bonding methods:1.

Your idea

Use a "tail" (presumably a separate piece of wire) to connect the equipment grounding conductor (ground wire coming into the box) to the bonding screw of the metal box.2.

The method being used

Taking the incoming ground wire, stripping a section of insulation in the middle of its run without cutting the wire, and wrapping the exposed wire around the bonding screw in the box, then continuing the wire to its final termination (typically a ground bar).

Code Analysis and Explanation

Your idea is much better than the others. The only problem is that you would need to be a UL-Listed grounding lug. This would allow the single conductor to be connected. NEC 250.8 Connection of Grounding and Bonding Equipment. (A) Permitted Methods. Grounding conductors and bonding jumpers shall be connected by exothermic welding, listed pressure connectors, listed clamps, listed means of connection, or other listed fittings. Solder is not an acceptable method. You have to use Listed/Approved products. I would recommend something such as a Polaris Tap Connector. These a liquid tight insulation tap.

The method being used (stripping and wrapping) is definitely a violation. Here's why: Lack of Listed Connector

The act of stripping the insulation and wrapping the conductor around the screw does not create a listed connection. The NEC requires grounding and bonding connections to be made with listed and identified means. Wrapping the conductor doesn't meet that requirement.

Potential for Looseness and Corrosion

This "stripped and wrapped" method is highly susceptible to loosening over time due to vibration, temperature changes, or even just the inherent pressure of the screw on the wire. It's also prone to corrosion because the exposed copper is vulnerable to the environment within the electrical box.

Potential for Reduced Conductivity

The contact area between the stripped wire and the screw is often poor. Corrosion makes this worse. This can significantly reduce the effectiveness of the ground fault path.

NEC 110.3(B): Installation and Use

Listed or labeled equipment shall be installed and used in accordance with any instructions included in the listing or labeling.

Why is proper bonding important? Safety

The main purpose of bonding is to create a low-impedance path for fault current to return to the source (transformer, generator, etc.). This allows overcurrent protective devices (breakers or fuses) to quickly trip and clear a fault, preventing dangerous voltage from appearing on metal parts of equipment that could be touched. Poor bonding can delay or prevent a breaker from tripping, leading to shock hazards and fire risks.*

Code Compliance

Proper bonding is a fundamental requirement of the NEC. Electrical installations must adhere to these rules to ensure safety and prevent problems.

In summary: The method being used by the people you work with is not code-compliant and potentially dangerous. If you would like you could provide the size and amount of wires that need to be connected to each other and i can find the exact part you need to be up to code. Flag for review