We have a Downtube FSIX and FSVIII. Both bikes have had flats from spokes coming through the rim strips into the tubes. I have since replaced all of the rim strips with good cloth strips and have had no problems since. We love these bikes, but this rim situation looks like something should be taken care of at the factory on new bikes.
Cloth rim tapes work much better than the rubber/plastic one's most bikes come with.
While you replace the rim strip check for extra long spokes, burrs and sharp edges where the spoke holes are drilled through the rim. Spin the wheel around with the tire, tube and rim strip off and listen for tiny noises which might indicate the presence of shavings inside the rim. Not uncommon for alloy rims and a potential source of surprise flats...
Bike Friday, Bridgestone MB-6 700c, Ti-frame Xtracycle, RANS, Brompton, Dahon, Downtube IXFS, ex-Birdy & a recumbent pedicab.
Rim tape lasts forever, provided you don't have sharp spokes or burrs constantly rubbing again it (rare) - that's why I'd do a quick sanding first just to make sure. Rub a finger around the rim; anything sharp or bumpy just file/sand/rub down. Other than that, the only time you may need to remove the tape is if you're replacing a spoke or something.
The only other thing is to buy the right sized width tape. For a 305 rim, you'll have plenty of tape left over. Maybe you'll be able to use one roll for two wheels. No guarantee though.