First let me say that ANY winch cleaning and lubing, unless done incorrectly, is probably far better than the thousands of boaters who go years without servicing a winch. But....
For my expensive $$$$ Lewmar winches I use, what else, Lewmar grease. I still have the same tube I bought about six years ago for about $12.00.
I suppose if you could find a calcium sulphonate grease for less money it would work just as well as the factory Lewmar grease but it seems foolish to even search for it when I know the Lewmar grease works so well, is easily available and a $12.00 tube lasts for many years. So far it has cost me perhaps $2.00 per year for winch maintenance and my tube is still about a third full... If I had Harken winches I'd buy Harken's suggested grease.
When my bell hosing cracked at 2800 hours on my Westerbeke, and I was trying to figure out why, their first question was "are the motor mounts Westerbeke?" to which I replied "yes". Long story short is they sent me a brand new bell housing, at no charge, because they decided it was an underlying manufacturing defect. I am going to guess that if I had skimped and saved a few bucks on aftermarket motor mounts Westerbeke never would have sent me this very expensive part at no charge.
Despite the price I like to stick with what the manufacturer recommends. I also use 303 Fabric Guard on Sunbrella and IMAR products on my Strataglass and AwlCare on my Awlgrip. These are all manufacturer recommended products that have proven themselves to me over the long haul.
Not saying the grease used won't work but Lewmar has a specific type of grease they recommend, that is reasonable price wise, and easy to get.. As others have said a light coating is all that is necessary..