I ended up fixing the issue by editing the scrambleafricascenario/turnsremaining.lua file as suggested here; I'm summarizing the solution below:
Use a text editor like notepad to open the "turnsremaining.lua" file,
located at SteamApps/common/Sid Meier's Civilization
Find the line below and change 100 to whatever number of maximum turns
you want for the game (tested OK with 300):
local iTurnsRemaining = 100 - Game.GetGameTurn();
Next, find the line below and change g_kiGameTurnLength to the same
value you inserted on the previous line:
local iTurnsRemaining = g_kiGameTurnLength - Game.GetGameTurn();
After this, the "Just one more turn" button will be enabled in the Scramble for Africa scenario, even for an ongoing saved game.
Also, bmaupin's Save Game Editor allows you to change several settings of a saved game, namely enabling the Culture UI, see the following for more detail: Is there any way to change victory conditions with the game already started?
There is actually a web version to make it easy to edit the saved game: https://bmaupin.github.io/civ5save-editor/