Congrats on the new home purchase! I also bought a new home last year and at that time, I purchased the x590 due to slopes in my yard. I am now in the process of trading it in for an x758 as the 4WD will very helpful and I also plan to plow my driveway during the winter. I got a fair trade from the dealer for the x590, but I would have been in a better position if I just purchased what I really wanted from the start.
I guess my point is that you should really consider all options before dismissing a possible upgrade. Will you just be mowing or is there the possibility that you'd like to remove snow and do other tasks around the yard with the tractor?
I know you said the x570 is out of your price range. If you do the math, the x570 MSRP before tax is $6k. Deere is offering loans at a 4.9% interest rate on the x500/x300 series. This comes out to about $138/month before tax, and I'll bet you could get the x570 below MSRP. The x380 is $4,399 and that works out to about $102/month. So, for about $36 more per month, you've got a much more capable machine. Again, these are estimates are before taxes.
In my opinion, the x570 would be more than enough to mow your yard and the hill shown in the photos. I would hate to see you buy the x380 and regret the decision later. If you're set on the x380, you can always try it and if you don't like it, you can return it on the JD promise within 30 days. I would just be concerned that you would change your opinion after the 30 day period, like I did. Here is some information on the JD promise...
https://www.deere.com/assets/pdfs/c.../warranty-protection-plans/res_jd_promise.pdf
Again, congrats on the new home and good luck with your new tractor purchase!