Whatever your decision for these drives "today", I recommend that you seriously consider retiring them at some point in the next few months. You're already worried that they might not survive a power recycle, so I say it's time to give up a few lunches and make this happen.
Just one or two new 2.5" external USB 3.0 drives will be faster, have more capacity, and won't require external AC power. Especially that last point, because I know of old USB-to-PATA enclosures that have a loud little fan inside, always using AC power even if the hard drive within has parked itself.
And if you're plugging them into a USB 3.0 port, all the faster, but they'll work in USB 2.0 if your hardware is older.