Hi Judy, I'm a former full-time Realtor.
First, do deals under 20K each exist in your market? What is the average market price?
I will tell you when agents got leads for homes WAY under the market average at my brokerage they would usually refer it out to someone else. Like others have said, it was the hungry agents who would bite. One agent who got such a referral said bluntly she doubted a deal would come out of it since there is no inventory.
Having said that, I do not think your expectations are too high or unreasonable depending on your market and what the average home price is. You will need to do a lot more research though than just asking on Zillow. Those sites are basically playing roulette for both clients and Realtors. Think of it like online dating, you'll need to cast a much wider net. Also, word of mouth is the best way to find your agent.
Do you attend any investor meet-ups? Do you have an insurance agent or another self-employed business owner who could refer you to such an agent? If you are part of an investment/real estate/wealth building community that's where you'll get your leads and referrals.
I know agents who specifically work with investors, low-balling offers (which is a HUGE no-no in this market.) To the point listing agents have complained to them about unrealistic offers and "wasting their time". They still did it though because it was part of their own business strategy (VOLUME) and it paid off.
So those kind of agents are out there but you'll have to look.
Also, personally I would not ask the listing agent to double-end the deal and represent you. It's always in your best interest to have your own dedicated and separate agent. This is what I tell all buyers. The reason why you are getting advice to have the listing agent make the offer for you is in the scenario you literally have no other options because no legitimate buyer's agent will represent you.
I think you do, but again, you'll have to dig. If you don't or can't dig, then go with the listing agent. You can also do online only deals through Open Listings which specializes in distressed properties.
Good luck and let us know if you ever find someone!