Where did you get that information from? Id tech is a clear engine family/line, they aren't named 1, 2, 3, 4, 5 and 6 for fun(except for the first one which really doesn't fit the rest).

1. Carmack himself says he wrote D3 renderer inside Q3 in C and then ported it to C++ (Doom 3 was first C++ id game).
2. Carmack himself says between Q1, 2 and 3 the move was "evolutionary" and lots of code stayed the same.
3. On Mac the main of D3 is even called quakeMain.
4. Radiant - GtkRadiant is shared between many games and various radiants (Net, Dark, Gtk, etc.) share a lot of code with each other, DarkRadiant is for example a fork of GtkRadiant meant for D3 and TDM.
5. The MegaTexturing code must definitely be shared between 4, 5 and 6 because it'd be crazy to reinvent this from scratch each time to do the same thing.

The only big/throw everything away jumps are from previous odd games (Commander Keen which was 2D, Wolf3D which was a super simple ray caster) to id tech 1 (Doom 1 and 2, pure software, fake 3D) to id tech 2 (Q1 and 2 and beyond, GL but often with software fallback, real 3D). And that kind of jump is like giving web devs shit for not writing Java browser applets and thus "not reusing what they made before". From 2 and beyond it's evolutionary with the data formats, editors, code, etc. and never a total rewrite from scratch. Case in point: the use of OpenGL (or wrapping other APIs in a GL like one when there is no GL), Carmack used to (90s) prefer it over DirectX but much later in life (2010s) switched his preference but the code stayed with OpenGL out of inertia. Even Rage and new Doom use OpenGL (id tech 5 and 6 respectively). Why would this happen instead of switching to DX if they threw everything away with each new engine and never reused code?

Granted: lots of stuff and the ENTIRE scripting system (and Carmack's opinion of scripting) change almost each iteration but "change most of stuff but it's very evolutionary and lots of code is same" is far from your claimed "don't even reuse anything".

As for his opinion on writing your own engine it's not super clear, few years back he admitted that Epic dominated that field, but he was happy to stop licensing his tech to outsiders because it was a burden and started almost by an accident ("It's interesting when you look at our technology licensing -- it was never really a business that I wanted to be in. In the very early days, people would pester us, and we'd just throw out some ridiculous terms, and we were surprised when people started taking us up on it."). He is also reportedly a very calm, reasonable and polite person so even if he thinks reusing engines is shit he won't tell us. One interviewer who interviewed him said he just turned around in his chair, didn't even say 'hi', answered all questions in length in detail with no BS and then just thanked for the interview and turned back away to his screen without a 'bye'. Embodiment of pragmatism and niceness, despite the lack of 'hi' and 'bye'.

Sources:
http://fabiensanglard.net/doom3/renderer.php
http://fabiensanglard.net/doom3/interviews.php
http://icculus.org/gtkradiant/developers.html
https://en.wikipedia.org/wiki/Id_Tech_6
http://www.gamasutra.com/view/news/125324/E3_ids_Carmack_Wil...