There is no one answer, because "the Americas" is an extremely large and diverse part of the world with many different countries being involved in its historical development over the centuries.

I understand that an argument can be made that the British empire had a larger impact

In Central and South America? Certainly not. The Spanish absolutely had the greatest impact here, given that these regions speak Spanish and practice Catholicism, as well as have considerable Spanish heritage (the Hispanic ethnicity is the direct result of Spanish and native mixing). It's hard to argue that any other country had a greater impact on Central and South America when the Spanish literally created a new ethnicity there that dominates the continent in the present day. The Spanish impact there is arguably even greater than the British impact on North America.

Spanish expansion with the conquistadors obviously ignited a fire of imperialism but I think it can be argued that the British wouldn’t have even tried to expand that far until they learned of the exploits of the Spanish

British colonial policy in the 16th century begun not as an attempt to copy the Spanish in their large-scale settlement of land, but simply seizing the gold and treasure carried on their ships between the Americas and Spain. The English were primarily more interested in profit-based colonial expansion, i.e. seizing resource-rich islands in the Caribbean that could produce cash crops. Only in the early 17th century did the English (or British, more accurately at this point) begin emulating the Spanish model of European settlement in the Americas with the first colonies in Virginia, but a key difference was the interaction with the locals: unlike the Spanish, the British did not engage in miscegenation with the Native Americans on any large-scale, and instead opted for a policy of segregation and displacement rather than assimilation. The history of North America may have been vastly different had the British taken the Spanish route of colonisation, with the possibility of a new hybrid ethnicity of British and Native American mixture.