Your question has a few different questions in it, starting with

Would this have been possible?

The answer is yes, although depending on where Washington was it might be more or less difficult. For example, over the course of doing research about a prominent resident of my hometown, I found a letter he wrote to Washington in August of 1776. I believe Washington would have been in New York around that time. He's actually refusing to do a favor for Washington, ostensibly on account of his not being the best man for the job (although he was also a loyalist who had a somewhat dim view of Washington on the whole, so that may have been more the reason for the refusal).

So, anyway, it was possible for Washington to send and receive mail.

Could someone just sen George Washington a letter and have it delivered?

Eliza Schuyler/Hamilton was not just "someone," she was the daughter of a prominent general in the Continental Army and the wife of a prominent founding father. So this isn't a case of someone just sending along a letter to Washington giving her take on what he should be doing.

In fact, her husband was on Washington's staff, and they were therefore stationed in the same area for a good part of the war. She befriended Martha Washington, and they have documented correspondence between themselves.

I unfortunately don't know the answer to your final question,

Did relatively "famous" people deal with spam letters from armchair generals?

but it is well-documented that people would try to visit and engage with famous people. Take Felix Mendelssohn's visit to Scotland (a little later than the time period in question), where he tried to meet up with Sir Walter Scott:

We found Sir Walter in the act of leaving Abbotsford, stared at him like fools, drove eighty miles and lost a day for the sake of at best one half-hour of superficial conversation.

Not a spam letter, but it was easy enough to find his address and head over to where he lived. Usually, if you could get a letter of introduction from someone known to the famous person in question, it would be a little easier, but obviously people like Mendelssohn tried regardless!

May I ask a follow up? There is a scene in HBO's John Adams where Washington agrees to take letters from Abigail Adams and have them passed onto his personal squire/mailman for safe delivery. Is there any evidence that such an encounter really occurred? I know that Washington had a favourable opinion of Mrs Adams, but the idea that he made considerations over her letters to John whilst he was facing the biggest undertaking of his life seemed far fetched.

Furher followup- In the earlier years, postage could be paid by the recipient. Are there cases of people for whom so much mail was received is recorded as being a serious burden? Would the postal service come after a recipient who did not postage? If somebody received 'hundreds' if not thousands of mail a day, that couldn't be cheap. Similarly, was a local postmaster held in anyway responsible if recipients refused to pick up their mail in order to avoid paying postage? I know many of those positions were patronage ones anyway, but still.