There's not a phase out of Oyster.
I doubt c2c do many transactions at ticket windows on Oyster so the cost of maintaining compatibility of the ticketing equipment with Oyster is probably not worth it.
One potential issue concerns season tickets. If you hold a Travelcard season issued on Oyster and wish to changeover to a National Rail point-to-point season then this can be done if was purchased and loaded via a National Rail ticket office. This will be done under National Rail changeover process so at a pro-rata rate with no admin fee. If the Travelcard season was purchased an loaded via London Underground then it's a refund and re-purchase under the appropriate refund rules so not pro-rata and therefore more expensive.
The diminishing number of National Rail ticket offices that deal with Oyster card transactions means this could become a larger problem, something that
@LonTravelWatch should potentially look into.