Great question! I think writing a chess engine stresses some of the important but oft over look skill sets. For example, performance is always top of mind when designing a chess engine. Since hardware has become so cheap and highly performant, there are many projects where performance is simply not a big concern (until the project garners significant traffic). So to answer you question, no, it doesn't cover everything. But it's a great training ground. Ultimatlely, writing actual websites will be the fastest way to see how a webiste works end to end.