Re: Garcia Mitchell 306
That eggbeater looks like it's in good shape! Nice find! They were designed for mono line, not braid, but original fused 20lb test Fireline should be OK on it. If you're going that way with it, tie a 10 yard length of 17 or 20lb test mono to the spool, wind it on tight, then use an albright or uni-uni knot to splice the braid on to it.
And yes, you can pack the gear box with marine grease, but don't pack it too tight or it will push out through the seals.