I am in the same boat with my 305 not recognized by garmin connect. spent a lot of time on interweb looking for solutions, tried lots of different things (resets, uninstalling, installing garmin etc etc) but no joy.
previously I downloaded my 305 data onto fetcheveryone site through their garmin plugin but when the plugin was made defunct by garmin I had no choice than to use garmin connect directly and that is when I suffered with this problem. I am reliably informed that garmin has stopped supporting the necessary software to enable the 305 to download onto 32 bit modern pc/laptops though could be ok on older 16 bit units. that last bit is gobbledegook to me though that is what I was told though if that was true then I would have expected more 305 users to have the same issue.
what I haven't done is try garmin's helpline. they were very helpful last time my old garmin died.
i came up behind some elderly walkers on a trail run and thought they'd spotted me but clearly hadn't as one old duffer seemed shocked as i passed him. he then began to shout at me saying i should follow the 'countryside code' and make a loud noise to warn them of me. i shouted back something to do with victor meldrew and kept running.