If the “literally any modern website” part is a hard requirement, then I would suggest to go for a fork of modern mainstream browsers with strong privacy/adblocking features. The justification for this is that browser engines can only get so light in terms of complexity and still support modern sites, but sites themselves can be made to be less resource demanding on the browser by selectively blocking unwanted elements. Adblocking is the obvious, blocking unwanted JavaScript would likely be the next best bang for the buck, but even clearing the cache after each session can make the browser feel faster if your bottleneck is memory/cpu instead of the network.
If the “literally any modern website” part is a hard requirement, then I would suggest to go for a fork of modern mainstream browsers with strong privacy/adblocking features. The justification for this is that browser engines can only get so light in terms of complexity and still support modern sites, but sites themselves can be made to be less resource demanding on the browser by selectively blocking unwanted elements. Adblocking is the obvious, blocking unwanted JavaScript would likely be the next best bang for the buck, but even clearing the cache after each session can make the browser feel faster if your bottleneck is memory/cpu instead of the network.