Hello and welcome to the forum. You have come to the right place for a question like this.
I take it you're talking about a season ticket, or an Anytime/Off-Peak/Super Off-Peak ticket? The default rule is that National Rail tickets are valid on all National Rail operators' services, along all valid/"permitted" routes, unless a restriction to the contrary is shown on the ticket - condition 13.1 of the
National Rail Conditions of Travel (NRCoT) refers. The aforementioned Chelmsford to London Terminals fares are routed "Any Permitted", so they have no such restrictions and can be used on any NR operator.
Though it is controlled by TfL and branded as a quasi-Tube line, the Elizabeth line is also one of the 23 NR operators. It is listed as such in Appendix A on page 30 of the NRCoT, which precludes the usual argument to the contrary by TfL staff. So the NRCoT applies to your travel on the Elizabeth line*.
When travelling on a flexible/walk-up National Rail ticket such as the above, you can change trains at any intermediate station along a permitted route, and also start, finish or break your journey intermediately. So you're entirely valid to change to Elizabeth line services en-route, and you can also exit and subsequently re-enter at Shenfield/Ilford if you want. These stations are obviously along the route to London Liverpool Street. These rights are set out in condition 16.2 of the NRCoT.
In respect of Whitechapel, TfL likes to pretend that the Elizabeth line magically transforms into an Underground line west of Stratford, and thus that the Underground ticketing rules apply (which generally don't allow break of journey). But there is no basis for this policy; it is essentially just something they have made up because it suits their general operating/ticket model. As mentioned above, the Elizabeth line is part of the NR network and so all the same NRCoT provisions apply as when you are travelling on Greater Anglia, for example.
Unfortunately, you certainly aren't the first person to have problems using NR tickets at gatelines managed by TfL staff. I myself was given grief for using a valid NR ticket to exit at Whitechapel and was similarly told I "shouldn't do it again", though I was allowed to re-enter a few minutes later. TfL have a notable problem with rogue gateline staff who make up rules, and of course even diligent staff are let down by the training TfL gives, which is incorrect in some respects (as mentioned above). Every time I've complained I've been given apologies and platitudes, but I don't think TfL corporately understands the issue, let alone has any intention of addressing it.
So in essence, the GA staff were correct, but you are likely to encounter problems whenever you try to exercise your contractual rights at Whitechapel. It's up to you how to proceed; if it cost about the same, I'd probably just use PAYG Oyster/contactless for the relevant journeys to avoid the hassle, but if that cost materially more, I'd be looking to complain every time you have problems and escalate it as far as possible, until hopefully something gets done.
*The position is slightly different when you are using PAYG Oyster or contactless, but for season/single/return National Rail tickets it is as stated above.