It really depends on your research needs. Standard errors generated using replicate weights are considered to be the ‘gold standard’. However, as you’ve noted, standard errors derived from simple household weights are often quite similar. Many researchers choose to not use replicate weights due to the computational time required, but many others do use them, especially when precise standard errors are required.
I think it is fair to say that most researchers specify the household weight as either a probability weight or frequency weight when not using replicate weights. The household weight is generally specified as an iweight when using replicate weights due to some of them being negative. I think most researchers use a single stratum, but some do use STATEFIP.