I'm leery of refurbished computers, especially if they are cheaper builds. (IOW, a 4 year old used Thinkpad would be expected work, a 4 year old Ideapad, not so much)
Here are some available computers in that price range. I'm assuming if the major factor is price, the size isn't.
The Mevinim can argue these out
https://www.officedepot.com/a/products/9304611/ASUS-VivoBook-15-F512DA-DB34-Laptop/ $329 office depot store pickup
Asus 15 inch. Ryzen 3 3250u 8gb ram
https://deals.dell.com/en-us/mpp/productdetail/8dzg $328.30 (dell, before codes)
Inspiron 3000 15 inch. Ryzen 3 3250u 4gb ram
https://www.walmart.com/ip/Gateway-15-6-FHD-Ultra-Slim-Notebook-AMD-Ryzen-5-3450U-8GB-RAM-256GB-SSD-Tuned-THX-Audio-Fingerprint-Scanner-1MP-Webcam-HDMI-Cortana-Windows-10-Home-/103459429?selected=true $350
Gateway 15 inch. Ryzen 5 3450u 8gb ram
https://www.walmart.com/ip/Lenovo-IdeaPad-Flex-5-14-FHD-Convertible-AMD-Ryzen-3-4300U-4GB-RAM-128GB-SSD-Graphite-Grey-Windows-10S-81X2000HUS/829136096 $369
Lenovo Flex 5 14inch. ryzen 3 4300u, 4 gb ram
https://deals.dell.com/en-us/mpp/productdetail/8dzi $401.79 (dell, before codes)
Inspiron 3000 15 inch. i3 11th gen 8gb ram
https://www.dell.com/en-us/member/shop/cty/pdp/spd/inspiron-14-5402-laptop/nn5402ejnwh?cartitemid=068600ae-3450-4cd5-98ef-152f034f3981&ref=carconfigedittitle&configurationid=a80dc987-06ff-4639-8ab2-bd8c8051c8bc $416.50 (dell, before codes)
inspiron 5000 14 inch. i3 11th gen, 4gb ram
https://www.walmart.com/ip/Gateway-14-1-FHD-Ultra-Slim-Notebook-Intel-Core-i5-1035G1-16GB-RAM-256GB-SSD-Tuned-THX-Audio-Fingerprint-Scanner-Webcam-HDMI-Cortana-Windows-10-Home/828696636 $429
Gateway 14inch, i5 10th gen 16gb
For comparison sake, the CPUs mentioned here are roughly in this order, weaker to stronger Ryzen 3 3250u < Ryzen 5 3450u < i3 11 < i5 10 < Ryzen 3 4300u.
ram speeds arent as important as total ram, so I didn't mention. Flex 5 and Inspiron 5000 are mid level builds, the other seem to be cheaper.
I think I'd go with the i5 Gateway (just buy a mouse, they say the trackpads aren't great) or the 8gb ram dell. The office depot one is good too, if in stock near you.
Return if it's not adequate......
Or bump your budget to $589 and get this
https://www.amazon.com/gp/product/B086226DDB . I'm currently using the intel version of it, very satisfied, though screen is dim.
or $749.70 (before codes) for a great machine, see DDMS psot
https://www.dell.com/en-us/member/shop/dell-laptops/inspiron-14-laptop/spd/inspiron-14-5402-laptop/nn5402eqsls?view=configurations&configurationid=80c930e9-11f2-4f5c-be78-e0c6b56d0593