Are pilots legally required to obtain a weather briefing prior to some or all flights?
No weather briefing is required.
According to the NTSB, the simple answer is NO (you aren't legally required to call the FSS). The NTSB states... Part 91 regulations do not specifically require the use of any particular sources of weather information for GA pilots, but do require that all pilots familiarize themselves with weather and weather forecast information before beginning a flight. Source.
Yes, at least for flights that are not in the "vicinity" of the airport.
Each pilot in command shall, before beginning a flight, become familiar with all available information concerning that flight. This information must include--
(a) For a flight under IFR or a flight not in the vicinity of an airport, weather reports and forecasts, fuel requirements, alternatives available if the planned flight cannot be completed, and any known traffic delays of which the pilot in command has been advised by ATC;
You are not required to call Flight Services for a weather brief, but you are required to get the weather reports (which can be called a personal brief). If you do all flight services and happen to have an accident, then your accident will contain the details about your call.
Most airlines nowadays only have briefings at their base/hubs. On the line stations they would probably get handed a 'briefing package' consisting of an Operational flight plan, Weather, Notams (and company notams), charts (pressure and volcanic ash etc), this would be printed out by the station staff or handling agent. The station staff will add more info like the special load notification, Dangerous goods notification. In some cases like for short hops the station may print out the actual weather (METAR) at the destination and alternates.