A few thoughts:
1. The 470 or V8 4Runners a few folks mentioned are great budget family 4x4s, with the following exceptions: 1) The timing belt needs to be replaced every 100k, and this is job usually costs north of $1K for a shop to do, so make sure you get one where the belt was recently done; 2) nearly all of the 4.7 V8s end up having the exhaust manifolds crack, leading to a annoying exhaust leak that can't be repaired without replacing the manifolds and cats (easily $2K+ job); and 3) when the starter goes out on a 4.7 V8, it is located beneath the intake manifold and is an expensive repair based on labor (probably $1K+). So, you WILL need all three of those jobs at some point in a 10-year ownership cycle, and unless you have good DIY skills, the total repair cost can easily be $4K+. I have personally already addressed issues 2) and 3) in my GX, as a DIYer (the timing belt was done just before purchase), but both jobs are hard and should not be attempted unless you have a lot of DIY auto repair experience.
2. If you want a true midsize 4x4 SUV but want to skip issues 1) - 3), get a 2003-2009 4Runner, but with a 4.0 V6 instead of a 4.7 V8. They use a timing chain, don't have the exhaust manifold cracking issue as bad as the V8, and the starter is located outside of the engine. However, it is a less powerful, less refined engine.
3. Being that you are on a strict budget and need a capable family vehicle, I'd actually recommend a 2013+ Subaru Outback instead of a GX or 4Runner, with the newer FB25 engine. You should be able to get one easily in that price range, they are very capable on muddy/snowy roads with good tires, and have plenty of family room. Fuel economy is much improved (ours will hit 30+ on the highway when driven right), they have few major issues, and are easy to work as a DIY or at a shop when something goes wrong. However, they are much less refined/quiet/comfortable compared to a GX, but they will cost much less to purchase, operate, and maintain over a 10-year lifecycle (fuel cost will be around 40% cheaper than a GX, tires are cheaper, etc). You cannot beat an Outback for a capable everything family vehicle on a budget.