So , in short I should have opted for the A7R IV.
Bummer.
It is a little tricky to navigate this area. If we skip the speculations to why it is like it is and stick to the facts your options to get AF on your old A-mount screw drive lenses on E-mount FF bodys are:
Sony LA-EA5 with A1 or A7RIV. This combo will use the sensor to AF and give very good precision due to that. Also you can AF almost over the whole image area.
Sony LA-EA4 with A7III, A7RII, A7RIII, A7RIV, A9, A9II (+ several older bodys as well). This combo will use the built in AF-module that sits above the semi transparent mirror (which always takes 1/3 of stop of light) in the LA-EA4 adapter. You will get the AF precision from an older A-mount body with the usual need for micro focus adjustments. The LA-EA4 also only has some central AF-points.
LA-EA4r from the company MonsterAdapter, this is a conversion kit for the LA-EA4 where one takes out the semi transparent mirror and the Sony circuit board and replaces with only a new circuit board from MonsterAdapter. It will after this conversion use the sensor for AF and gives AF with many screw drive lenses and many different bodys. I haven't found any official page for this kit, MonsterAdapter is a small Chinese company but there is quite much info out on the net if you Google search it.
A note: The Sony LA-EA3 adapter works on many/most bodys but only with A-mount lenses that has built in autofocus motors. Both the LA-EA4 and LA-EA5 has this functionality built in so basically you get a "free" LA-EA3 with the 4 and 5 adapters.
The LA-EA2 is an APS-C version (the inside baffle is narrower) of the LA-EA4 adapter.
And the LA-EA1 is an APS-C version (the inside baffle is narrower) of the LA-EA3 adapter.
The best way to be sure if a combination works is to use the Sony compatibility matrix they have on their official site. Also the A-mount and E-mount forums on DPR, FredMiranda and
https://www.dyxum.com are good sources for finding out more specific information and how the different adapters works with 3d party A-mount glass etc.