And who is going to issue you a reasonably-priced intermediate cert? Especially since PKIX name constraints don't actually work, so that intermediate cert would let you sign just about anything.
For your own web servers and your own clients, use your own CA and your own certs.
edit: basically, what stops you from doing this?
Root CA root cert (20y expiration)
v
Root CA intermediate cert (10y expiration)
v
Your intermediate cert (1y expiration)
v
Your short-lived cert (24h expiration)