I didn’t finish the hand route, just did enough to see that it is probably possible (a lot better than the autorouter!) You do need to set DRC down to 0.05in as the 328 part has fine pitch pads and fails DRC with the default settings.
click on Router->Auto routing/DRC settings and select custom and set keepout from 0.01in to 0.005in like this.
After you start autorouter you can increase the number of rounds and it will remember the setting for later runs like this.
either type a new number or use the increment/decrement bars on the right side to increase or decrease the number of loops.
Again the 328 part is small pads and the default 24mil trace is too large for it. As well larger traces take up more board area and make routing more difficult (both manually and for the autorouter!)
I haven’t used the really high end software (Pro engineer at +$100,000 per seat is one!) but I understand autorouting (which is a hard problem!) doesn’t work all that well there either. They may let autoroute do some of the work but I expect they finish it off manually. Human eyes are much better at seeing possible routes than any software I know of.
A few additional notes, you look to be missing some things. There are usually two capacitors on the crystal (from the 328 data sheet):
which are not present. There are also usually power bypass capacitors close to the VCC and ground pins on the micro, typically a 0.1uf ceramic cap for hs noise and a tantalum electrolytic for current surges although they appear to be recommending 10nf ceramic bypass caps. You likely also need both a reset and programming circuit (the spi may be the programming interface though) and I don’t see either of those. This paper
may be helpful about what is needed, as it covers all these issues. See the drawings on page 7 for reset and page20 and 21 for pcb layout examples for power decoupling and crystal layout.