Is it hard to find a big random easy to factor number? Suppose that I give you the challenge of successfully factoring any very big random number. That is, you pick a big random number (say, 65536 bits) and try to factor it. If you manage to, you win. If you don't, you can keep trying, or pick another random number, until you factor any big number. How hard is to win this challenge?

To put it in another way, suppose that I offer you a lot of money if you give me a tuple (nonce, rnd, factors), where nonce is any number selected by you, rnd is the 65536-bit variable length SHA3 output of the nonce, and factors is the list of prime factors of rnd. Will you be able to take my money?
Yes, this is feasible. Just generate sufficiently many nonce, rnd values, and you will eventually stumble upon a prime (or a number that can be factored into a large prime and a number of small prime factors). This is how most probabilistic prime generators operate.