Grounding the strings makes good sense for reducing pickup noise, but it does give a very low resistance path to ground for your body if you touch anything that carries mains voltage. Several people have been killed and many more badly shocked when a faulty PA system sent the mic live. Touching the mic whilst holding the guitar formed a very good electrical circuit, and often if there was a hand on the mic and a hand on the guitar, the current would flow straight through the heart.
The normal ground path through your socks/tights shoes and stage is has a pretty high resistance, but the low resistance ground through the grounded guitar strings allows for a much higher current to flow, and it's current that kills, not voltage.
The days of valve PAs have (unless you use retro equipment) long gone, so the risk of a high voltage getting to the mic is almost negligible now, and the use of RCDs/ELCBs should cut off power before any real damage is done if it does happen, makes things a lot safer. But people still do stupid things like remove safety grounds from amps to cure ground loop hums, instead of doing it to connecting cable shields or using isolating boxes, so the risk is never zero.
With low impedance active pickups like EMGs, their noise pickup level is so low that grounding the strings (and allowing your body to be grounded by touching them) has no benefit, so to improve safety they say not to ground the strings.
Of course if you don't ever play and use a mic, then you are pretty safe. Using a radio lead with a mic (or a radio mic with a guitar) is also a lot safer than using a wired mic and guitar. But you only have safety issues with faulty or damaged equipment, or badly installed wiring (more prevalent in older buildings).
The tailpiece grounding will work just as well, as long as you can drill a hole to it. It's how my ES-3 strings are grounded.
Otherwise, I'd pull out the bushing, drill a hole and do it properly. If I remember correctly, you've pulled out a bushing before with a puller, or else there's the old 'end cut off a smaller bolt, dropped down the hole and an M8 screw turned in the bushing to push the bushing out' trick.