Does this mean a fee of greater than 0.01/kB wouldn't increase the 1st confirmation speed?
In most cases, yes. If your fee is enough to get all miners to include your transaction, then there's no point in adding more.
The network increases the minimum fee automatically when there are many transactions queued. Nowadays the default minimum fee is lower, though, so 0.01/kB will usually be sufficient even when the network is being spammed.