Every opamp has its allowed "input common-mode voltage range" listed on its datasheet. The TL084 has an allowed input common-mode voltage range that is from 4V above its negative supply voltage (an allowed input of at least +4V in your circuit) to 4V less than its positive supply voltage. But many of them have inputs that work as high as the positive supply voltage.
The TL081 is an old opamp with Jfet inputs. It has a problem called "Opamp phase Inversion" where the output suddenly goes high if an input voltage is outside the allowed input common-mode voltage range and your input voltages are too low so of course the output latches high.
The OP07 has an allowed input common-mode voltage range that is from 2V above its negative supply voltage (an allowed input of at least +2V in your circuit) to 2V less than its positive supply voltage. Again, your circuit will not work with it.
There are many opamps that have an allowed input common-mode voltage range that includes their negative supply voltage (0V in your circuit) like an MC34071 or a TLE2141.