Why do scientists usually measure the initial rate of a reaction, instead of measuring the rate some time into the reaction?
First, note that one does not always measure initial rates. It is fairly common to fit an entire reaction curve (or nearly entire), but it requires a little bit more math. It is also common to measure an "initial" rate that is not quite initial. That is, it is the rate after a burst phase has occurred, and the system has settled into something close to a steady state.
That said, measuring initial rates is done for two practical reasons.
1) we can approximate the product concentration as 0. For a reaction that is at all reversible or product-inhibited, the math is much easier if [P]=0.
2) we can often approximate the concentration of one (or more) of the reactants as fixed. For example, if you measure the rate only for an amount of time required for a few percent of the reactant to be consumed, you can get a pretty good fit to the data by assuming that reactant is fixed at its initial concentration. That makes it easy to produce plots of the initial rate vs the concentration of that reactant.
Both of these factors make it much easier to determine basic rate laws and the approximate value of the associated rate constants.