Not at all a scientific comparison but I tried to create comparison groups for "greats" and "lesser lights" for the Open Championship and the US majors that overlap. My take is that better courses do tend to produce better champions than lesser lights.
I tried to pull the champions from 1900 and later:
Open Championship
Greats:
St. Andrews - Lois Oosthuizen; Tiger Woods (2X); John Daly; Nick Faldo; Seve Ballesteros; Jack Nicklaus (2X); Tony Lema, Kel Nagle, Bobby Locke; Peter Thomson; Sam Snead; Dick Burton; Denny Schute; Bobby Jones; Jock Hutchinson; James Braid (2X); JH Taylor.
Muirfield- Els, Faldo (2X), Watson, Trevino, Nicklaus, Player, Cotton, Alf Perry, Hagen, Ray, Braid (2X);
Lesser lights
Sandwich: Clarke, Curtis, Norman, Lyle, Rogers, Locke, Whitcombe, Cotton, Hagen (2X); Vardon, White
Troon: Hamilton, Leonard, Calcacecchia, Watson, Weiskopf, Palmer, Locke, Havers
US Open/PGA
Greats:
Oakmont – Cabrera, Els, Nelson, Miller, Nicklaus, Hogan, S. Parks, Armour, Mahaffey, Snead, Sarazen
Merion – Rose, Graham, Trevino, Hogan Dutra
Shinnecock Hills – Goosen, Pavin, Floyd
Lesser Lights
AAC – Pate, Nelson, Toms, Keegan Bradley
Hazeltine – Yang, Beem, Stewart, Jacklin
Medina #3 – Woods (2X); Irwin, Lou Graham, Middlecoff