'Use the Central line' is the advisory during engineering work because at weekends it can cope with that level of crowds (albeit only just, load factor is still >80% in the central sections for a lot of weekends). During weekday peaks it's 100% at capacity even when there is no disruption on itself or elsewhere. There is no room to divert any meaningful quantity of TfL passengers onto the Central during peaks, they'd have to switch to Anglia services (which is already advised during disruption / strike action) as they have some degree of headroom - 12-car Southend services, when not short formed, often have spare capacity for 400+ passengers each even after Stratford, let alone before. I believe this is less on other services, but there's a fair amount of extra room there, that's where the displaced passengers travelling between Stratford and Liverpool St would go - otherwise there are other diversionary routes that could potentially be used e.g. Jubilee to West Ham and District/HC, Jubilee directly into zone 1 or DLR to Poplar then to Bank etc. depending on where their destination is.
That, however, isn't the real issue here. The real problem is the actual capacity loss of TfL Rail itself - terminating at Stratford with a single pair of platforms is going to reduce the service interval to probably 8tph at best. Between Stratford and Ilford, the service is already full in the peaks. Where are all those passengers going to go? TfL won't be able to discriminate between passengers using the service to zone 3/4 destinations where there may be semi-viable alternatives on buses/district line/c2c and passengers travelling further afield. It will end up being necessary for people travelling as far west as Romford and Gidea Park to go to Shenfield on an Anglia service and come back. Not popular!