You are correct that if thrust is constant, and TAS increases through temperature then angle remains constant, and rate increases.
If you were talking rocket engined aircraft this would be usual.
To the gas turbine engine a rise in temperature causes a rise in power required to produce the same thrust;
T=mv, but kinetic energy is 1/2mv2.
A drop in air density with temperature increase, requires an increase in V to maintain thrust. But that is squared into power required, whilst the drop in mass is halved.
Then the TAS is increasing so the prop/fan/jet speed must again be increased, again requiring more power yet.
An increase in temperature causes an increase in power required, a reduction in power available, requires higher compressor speeds, brings you closer to your EGT limits.
So as far as a gas turbine goes you are comparing a reduced power climb on a cool day to a max power climb on a warm day.
