Hi @toment
If you are taking/considering an offer from EE for their (EE) broadband and/or mobile services you need to look at their (EE) offer terms, There is an EE community with members familar with their products , the TV section being
https://community.ee.co.uk/t5/TV/bd-p/TV
If you currently own your existing TV or TV recording box presumably with a satifactory working Freeview aerial connection you could subject to having sufficient HDMI ports on your TV continue to use them as well as for example any EETV box that comes with your EE package combination. ie one TV HDMI connection to your new EETV box another to your existing Freeview recorder and presumably one for your Firestick. One consideration would be that sometimes (particularly if your existing Freeview recorder is of Humax manufacture is that remote button presses might be recognised by both your existing recorder and some EETV boxes if in line of sight.) EETV do offer a recordable box (the Box Pro) as well as other options including Apple TV and Box Minis , the Box Pro will operate with an aerial connection , but also provide as an alternative instead to have IPmode with a selection of freetoair channels (some in Higher Quality than their Freeview equivalent) with some restrictions on adskipping for ITV and recording playback for BBC where it directs you to iPlayer..
As silly as it sounds currently the range of EETV products offered and available to BT Group customers differs depending upon whether EE or BT is the Broadband supplier.
Normally if you have an APP subscription billed (ie included in your package or addedon to it) by BT/EE you would be free to use that app on other devices such as your Firestick subject to the APP normal device and concurrent user limits. Normally if Netflix is included within an EETV package EE/BT would be anle to takeover the billing of your existing account but you might need to take an addon to your EETV pack if for example your desired Netflix subscription is a higher level that the subscription included with the EETV package. You may also need to consider NOW Boost addon if you intend to have concurrent viewing of NOW (entertainment) content on an EETV device and another device.
Thank you so much for your comprehensive answer- much appreciated.
Tom