Personally, I think ticket office staff should know about some common local ticket splits and advise customers towards them. Obviously it would be unrealistic to get them to learn every split ticketing combination, but local ones and then the major boundaries (to get PTE tickets etc.).
Err, no. It is strongly advised not to voluntarily offer splits. They confuse customers because you have to spend time explaining all about what to show and when, which ticket to put through which barriers, time restrictions, Condition 19c and so on. Most customers just want you to give them what they ask for and I think it's mutually beneficial if that is what is done. The only time I suggest split tickets is when I'm asked for;
a.) Outboundary Travelcards where customers haven't come from outside of London - they get an All Zones Travelcard + return extension combination.
b.) To confused customers who start from London, need to go beyond the zones and back in, then to somewhere else but haven't a clue what they need to buy. I.e someone asks for a day return to Gatwick, then are going to Colchester later on. I will give them an All Zones TC with the appropriate BZ6 extensions, it covers all the bases, usually the cheapest option and only warrants a very brief explanation. I have been known to do something like an SDR to Gatwick and an overdistance excess to Colchester for the return portion together, but only because the customer specifically asked for an extension on the return.
c.) A cheaper fare where the customer doesn't want a SOS/R. I will not offer anything which may require a lay passenger to argue their CoC rights to on board staff, in these situations I keep to a maximum of one split, where the customer has to change trains anyway. Up to 75% can be saved on single fares this was. I don't do this unless I'm specifically asked (customer may know that certain trains are cheaper than others and ask to use the cheaper one part way) and am not obliged to do it unless specifically requested.
d.) Where no through ticket is available, which occasionally happens.