While I agree it would be nice, I can imagine there would be spam related issues at play with that feature. Namely… you’d never know if your invoice was received.
A website can open your mail client for you, but I don’t know if it can attach a PDF for you.
Might be possible to set a download link in the body of an email via an href mailto:.... ?
Attaching the actual PDF might be another 'thing' that triggers some spam filters.
You should have a look at bizzey.com. I've switched 2 months ago from excel to this and it automates most of the tedious tasks of sending out invoices including checking if the payment arrived on your bank.