While doing some work for me, an electrician looked at the lug-type breaker panel and advised me to change the panel to include a main breaker. However, the meter is located significantly over 10 feet away from the panel. Several web sites suggest that a main breaker is only appropriate if the meter is within 10 feet of the panel. Why is that so?

The "10 foot Rule" for this application is not written down in the Code, anywheres.

It comes from the Tap Rules, that does not apply here.

The service entrance conductors that enter a building shall terminate at the " inside nearest the point of entrance of the service entrance conductors".

Many, (not all) inspectors, allow up to 10 feet of these conductors to pass through a building to get to the Main Service Disconnecting Means, AKA Main Breaker.

Consider this, these wires have NO short circuit protection, until they get to the Main Breaker.

If someone were to drive a nail into these wires, they will burn until something melts clear.

... and that is the only answer required.