For some teams, (and many teams in the Premier League), tickets will never reach General Sale. You’ll now have to consider if buying a Membership from the club makes sense. For some clubs like Tottenham and Manchester City, you’ll be required to buy a membership to get tickets, but as long as you look to buy them as soon as they go on sale, you’ll likely be successful getting them. Clubs sometimes have different tiers of memberships, which open up different ticket buying options, and usually come with some perks like discounts to the team store and small collectibles.
For high-demand clubs like Arsenal and Liverpool, simply getting a membership might not be enough. Generally, these tickets become available to members through seniority, which is earned through a combination of tenure and the number of times a member has bought tickets. You’ll find that for a US fan of a high-demand club hoping to buy tickets as a member, you might have to look for low demand games, like a Carabao Cup game, early-round FA Cup game, or Champions League/Europa League group stage game. I've had friends have success going this route, but you have to be content with trying again and again for tickets and then be flexible with your travel plans. It's much cheaper than Hospitality tickets, but more unpredictable.
Depending on the club, membership pricing on tickets is going to be well worth the cost of having to buy a membership, and as I mentioned, some larger clubs like Manchester City or Tottenham often have tickets reach members with little or no seniority. However, if the club or particular game you’re hoping to see is in too high demand, the purchase of your membership could be a sunk cost.