As I understand it, all of Tohorot has been reduced, in modern practice, to Niddah and some hand washing. The entire structure of transmittable 'uncleanliness' is inoperative. Why is it inoperative? Because there's no Temple, and so no ashes of the parah, to get anyone into a definitively tahor state?
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It's actually almost entirely still operative. People and utensils transmit and accept impurity just like they always did. What has changed is a lack of impetus to purify one's self in most instances. There are essentially three ways of overcoming a given impurity:
The laws that accompany Taharot have a number of applications to other areas of Halacha. Sechach for instance can not be a vessel which would accept Tumah. Produce in Yerushalayim must come in contact with a liquid before having Maaser Sheni seperated from it. Kosher Mikvaot are needed for vessels bought from non-Jews (this is not an impurity concern). Hand washing before bread as you mentioned has some related concerns, including what amount of food requires washing. Male Kohanim are prohibited from contracting corpse-related impurity (this has lots of applications). There are likely some other small points which I am not thinking of now.
In the meantime, by learning the Halachot of Taharot it should be considered as if we kept them.
A large part of your answer is already suggested in the question - since the purification process for contact with a corpse is not possible nowadays. Another important point is that even in the time of the Temple, observing a state of purity was only required in order to eat sacrificial/holy foods, and since we don't have these it's not as important to remain ritually pure.
[It's worth noting that even nowadays Cohanim do make sure not to become impure by coming in contact with dead].