Here are regular season games played from baseball-reference's Stathead... I used 5 foot, 10 inches as the dividing line.
All-Time (since 1871):
=71 inches, all-time: 14,748 players
= 2,000 games: 55 players; >=71 inches, >= 2,000 games: 191 players
= 1,000 games: 451 players; >=71 inches, >= 1,000 games: 1,171 players
Divisional Era (1969-2021):
=71 inches, total: 8,582 players
= 2,000 games: 13 players; >=71 inches, >= 2,000 games: 128 players
= 1,000 games: 133 players; >=71 inches, >= 1,000 games: 749 players
When analyzing longevity, great players don't always have the longest careers, but guys that stuck around the longest (and kept on rosters by clubs) were almost always all-time greats. On the "short" list, most of the Top 20 are Hall of Famers. For the taller men, the Top 30 or so were Cooperstown caliber. Most of the former played around a century ago through WWII, while the majority of the latter played in the past 50 years.