https://pcpartpicker.com/list/ysMgsk
CPU: AMD Ryzen 5 3600 - 194.79
You mentioned "onboard". Did you mean integrated graphics? The Ryzen 3200G and 3400G come with onboard graphics, but lack the performance of the 3600. With a 3600, you're going to need a GPU.
MB: MSI B450M PRO-M2 MAX Micro ATX - 78.98
Don't have to worry about BIOS Flashback with a MAX board, and you specified something compact.
RAM: G.Skill Aegis 16 GB (2 x 8 GB) DDR4-3000 - 64.99
Pros: Known name brand, Newegg reviews say it works with Ryzen. Cons: it's not on the MB's QVL list. May want more RAM for video editing.
SSD: Team L5 LITE 3D 480 GB 2.5" SSD - 53.99
2.5" SATA drive. You can go much cheaper or more expensive here depending on space and quality. If video becomes a focus, you will want a larger drive.
GPU: Gigabyte GeForce GT 1030 2 GB GDDR5 Low Profile - 89.99
It probably won't play Cyberpunk 2077 really well, but is a little better than onboard video. 2GB of VRAM will hinder video editing, though.
OR
GPU: Gigabyte Radeon RX 570 4 GB Gaming 4G Video Card - 139.95
This card can currently play Gears of War 5. If you buy it from Newegg, you can get Borderlands 3 or Ghost Recon as part of a promotion. Has 4GB of VRAM which will help with 1080p video editing.
Case: Rosewill FBM-06 Micro ATX Mini Tower Case - 28.98
One of the cheapest Micro ATX cases on PC Part Picker since you mentioned compact. Only comes with a single rear exhaust fan. Would need to spend a little more to get a better case. You can also get 120mm Arctic F12 case fans on Amazon for less than $8, and I recommend buying at least 1 extra fan for intake.
PSU: Corsair CXM (2015) 450 W 80+ Bronze Certified Semi-modular - 57.50
You don't want to cheap out on the PSU. The CXM is a decent semi-modular PSU which helps when building in smaller cases, and Corsair is a reliable brand.
With the RX 570, the total is 619.18 after MIR and shipping costs. You can save money and hit that $600 cap by going with the GT 1030 if gaming isn't a concern, and spend that money on a larger SSD or better case. But the cheaper GPU might not be good for video editing (not my area of expertise).
This build does not includes the OS.