74HC14 Schmitt Trigger Oscillator - Weird output at certain voltages
Check the function without LED.  It's well possible that your LED has too low resistance current limiting resistor which prevents the full output swing. This theory is compatible with the improved operation at lower supply voltages.The resistance in series with the LED should be at least 2 kOhm.

Addendum due the downvotes and comments:

At +5V supply voltage 74HC series output current sink or source capablity is rated to be max 5 mA. Loading that heavily or more is well visible as narrowed output voltage swing. Let's assume the led has 1,5 V forward drop. This leaves 3,5 V to series resistor. 2 kOhm series resistor determines the LED current to be = 1,75 mA. This is about a third part of the rating, surely on the safe side. If we want to stretch to the limit, then 680 Ohm series resistor is absolute minimum, but the oscillation frequency is difficult to pretend, it can be remarkably lower than without the output load.

As written in comments, a capacitor over the supply voltage, with short wires to the IC (=decoupling) is needed to keep the supply voltage stable. This is a basic practice in logic circuits. No decoupling implies parasitic oscillations (=unquessably complex instability).

Another basic practice is to short unused logic inputs to high or low. Open MOS inputs easily collect stray signals that drive the IC to have half-state logic elements which are unstable and draw plenty of current. On breadboard all IC pins are connected to metal strips that are substantial capacitors. That increases the stray-effect remarkably.

Reducing the supply voltage makes a logic IC slower, but it still can operate. Slowdown is caused by the increase of internal resistances which makes all time constats longer.  A slow IC is less prone to parasitic oscillations because its frequency bandwidth is low  when compared to normal.

By the questioner noticed "noise" can be high frequency parasitic oscillation. It can be quite chaotic, far from pure sinewave, but some short term periodicity should be seen, if the oscilloscope is set to have 0,1us/div.