Do battery strips prevent an iPhone's power from cutting out?
Battery strips are adhesive, stretchy "tape" strips. The purpose of them is to keep the battery in place, as well as slightly insulate the battery from the casing. The insulation means that sudden applications of heat or cold to the outside case (such as for example taking the phone out in intense sun light, or placing it on a very cold surface) doesn't immediately transfer the heat/cold to the battery via the aluminium casing.

If you're not exposing the phone to these changes in heat/cold when the cutout happens, it just sounds like you've received a non-Apple battery replacement of low quality.

Usually these cuts outs happen when the phone hardware uses more power than the battery can provide. As the power draw increases, the current (i.e. volts) are lowered slightly. With a bad battery, a normal power usage can lower the current so significantly that it is below the requirements of the CPU. This means that the system just shuts off immediately.

It is obviously not healthy for a phone.

I see it as very unlikely that new battery strips will solve your problem, unless you specifically see the problem when shaking the phone, or exposing it to heat/cold. Your problem could potentially be solved by replacing the battery - but then you'll also want battery strips installed ofcourse.