When you have missile on your tail, the engagement did go wrong in the first place. But I would rather have a missile on my tail in plane flying at 800 knots, then in one flying at 250 knots.

That is true, but besides that, two most impotant things you have to have for evading missile on your tail are high pitch rate and high AoA.
That means abrupt pitch maneuver.

Now why would some point the aircraft, if he has highly manouverable all-aspect, helemt mounted sight guided, IR missiles ??
.

Because, missiles are not allmighty. They are limited in flight time, engine burn time, max g, off-bore sight, etc.
When you read that some missile has 9 g capability, that doesn’t mean that it could be sustained. When engine stops, max g drops sharply, etc,etc.

For that reason, firing aircraft has to have good point and shot capability, to acquire firing solution faster.

When two missiles have similar off-boresight capability, the one which is launched at more favorable AoA will have a more effective range.

That is why, aircraft AoA is important.

How was the F-4 a poorly designed fighter?

What do you call the R-73 then? A substitute for poor BVR avionics and weapons?

The F-4 has bad aerodynamics. It is very sensitive to departure above 20 degrees AoA, and has high wing loading and corner velocity, but that is another story.

The modern fighter must have a high pitch rate and AoA capability to point the weapons and shoot first. If you have high off-bore sight missiles, that is added benefit.
It is no wonder why German Mig-29 beats everything put against it within 20 km in NATO exercises. It is too dangerous.

Delta configurations are not good for a fighter. The aspect ratio is too small and it effects badly on sustained turn, climbing turn etc. The induced drag is too high.

Regarding F-22 versus SU-27/Mig-29, in close combat F-22 has no chance with 60 degrees max AoA.
SU-27, with its tremendous pitch rate (60 deg/sec), and 120 degrees AoA, will point and shoot with R-73, first.

SU-37 has unlimited, 360 degrees AoA.!!!

The F-22 ,with its 40 degrees max pitch rate and 60 degrees max AoA, has no chance to fire first, in close combat.

Its only chance is firing from the distance with AMRAAM, but that could do F-4 also, or any other bad designed fighter. The AMRAAM is substitute for a bad aerodynamics.