IF you do take it apart to replace the spline gear, you should plan on using new seal for the center axle/spline and there are 1 or 2 (varies with year) seals on the rod which goes thru to operate the brakes. Plan on new 'wave' washers for going under the cover bolts around the outer edge too.
It sounds like someone was overly generous with the general purpose grease - at least they did have the thought to put something on there, even if it wasn't a great choice. What you want on those is an even "schmear" or coating (not big gobs) of a fairly sticky paste grease, preferably with a fair amount of moly in it.
There are a variety of brands available in the USofA - one of the most frequently used brands can be obtained here at Honda motorcycle shops (might have to be ordered, though) and is refered to as Honda Moly-60. It sticks to stuff well and is designed specifically for applications like spline lubrication.
IF t'were me, I'd clean up the brake shoes - replace them if too contaminated, clean everything up and put some moly paste on there with an old toothbrush and ride it. It does look like your splines are getting worn and you will have to plan on replacement
by year's end depending on how many miles you rack up. Once the splines get to a pointy knife edge they can strip/fail and spoil your day. The should have a more "square" profile, like this:

You can probably score a used final drive in good shape for less dosh than buying a new spline gear. I found a complete final drive with very good splines on ebay a while ago for about $125. Also, I have heard that some talented machine shops can repair/rebuild the splines by welding material back onto the teeth - I confess to never having seen this done, but some claim it is possible.
From what I can make out from your image, I think you have a bit of time yet, but I'd be planning for a repair/replacement either before too l