Well, yes. But if you're not keeping track of that, then always using 15/49 will give you marginally better results, on average, than always using 4/13. Perhaps there will be times where, if you had kept track of the cards, you would give odds of 4/13; but you didn't, so you don't know that's the case, and you should give 15/49.

You're correct about your other objections, and 15/49 is indeed harder to work with - but "without actually making it more accurate" is false under the one-deck assumption. It is not wholly accurate, but it is more accurate.