What should I do if I play tradition civ on large uncontested land?

I will agree with your first point and argue the second. Tradition does expand out quicker than Progress in the early game. However, its the infrastructure part that Progress shines in. Progress has more workers, more gold, more production, and more food for those satellite cities. Progress satellites gain infrastructure at a faster pace than Tradition ones do, that's its main strength.

More food? Do you forgot that Tradition has extra city growth for every city. From the starting policy, and growing more powerful with each policy, instead of a flat 3 food, in a later policy. Tradition secondary cities reliably out pace Progress cities in terms of population.
 
I see "Tradition has an earlier time going wide early" posted a lot, and I think it's a bit shallow of an analysis.

Tradition needs population in it's capital. If you use the three population to just spam settlers, you will be able to acquire land, however you will hurt your culture output substantially. The difference between 4, 5, and 6 culture at this stage of the game is huge. Your capital really needs to hover around 6 pop for 6 culture (2 from monument, 1 from palace, 3 from tradition) to roughly match progress in culture, anything below that and you are way slower. This gets worse if you want the engineer before the artist.

So yes, tradition can grab land quite easily, but it does bear an opportunity cost.
 
I see "Tradition has an earlier time going wide early" posted a lot, and I think it's a bit shallow of an analysis.

Tradition needs population in it's capital. If you use the three population to just spam settlers, you will be able to acquire land, however you will hurt your culture output substantially. The difference between 4, 5, and 6 culture at this stage of the game is huge. Your capital really needs to hover around 6 pop for 6 culture (2 from monument, 1 from palace, 3 from tradition) to roughly match progress in culture, anything below that and you are way slower. This gets worse if you want the engineer before the artist.

So yes, tradition can grab land quite easily, but it does bear an opportunity cost.

I don't disagree here, but I think Tradition has an easier time eating (quite literally) the cost of a settler than Progress does. Their growth bonuses mean you can make a settler, regain the lost population, make a settler etc. much easier than with Progress (especially since one of Progress' bonuses is based on capital growth *but does not proc if that growth has already been achieved*).
Progress also has troubles with settling early, namely the fact that your science output plummets with each city, which makes your culture output plummet.
 
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