I've been in your place before.  The truth is, any riding mower that cost a few hundred dollars used, or even $1200 new is a throw away.  Depending on how you use it, it might last several years.  If you didn't have 5 acres, I'd say a good riding mower is what you need.  
I'll tell you my story and you can compare.  I live on 3 acres in the country, and like many rural properties, its hilly, bumpy, I have trees, I handle fire wood, and I have a paved driveway that is 600 feet long to clear snow from in the winter.  I bought a $1000 riding mower for the first go around.  It lasted maybe 4 years and was falling apart, the deck was shot and so was the steering.  I sold what was left of it for $50.   I then bought a high priced riding mower, something like $4500.  It held up better, but all it did was mow, I got stuck every so often, and it sucked on my hills.  Like you say, one tire spins.  When I sold this $4500 mower after a few years, I got $750.  I now have a Kubota 
BX2350, end loader, rear blade, 60" mowing deck (mid mount mower) and a rear blade for the 3 point and a 4' 3 point brush mower.  Brand new, this stuff will set you back over $10,000, closer to $15,000.  Well used stuff will still be over $5000.  
I feel like it was money well spent, I don't hate mowing anymore, with a end loader, you won't use a shovel and wheel barrow very much, snow removal is so much easier, no more walking behind a snow blower.  I can move piles of dirt, gravel, dig holes, all while sitting down.  Its saved me a lot of back pain.
You have to think of this as an investment, a long term purchase.  Figure out how to save for it or afford the payements, and you won't regret it.