True replacement for MOSFET Infineon SPD04N50C3
The front page of the STD7NM60N data sheet states clearly that it is intended for: -


  They are therefore suitable for the most demanding high-efficiency
  converters.


This means that they ARE susceptible (almost certainly) to situations where the gate-source voltage isn't as robust as it could be. To verify this, the best place to look is the graph for ID versus gate voltage. Most MOSFETs have this graph and it tells you how susceptible the device might be when operating from a non-ideal gate-source voltage.

Of course, if the MOSFET is designed for switching converters then this graph is of little consequence because it is always assumed that the gate drive voltage will be around +10 volts and well-above the (circa) 5 volt area that can cause thermal runaway (yes, MOSFETs do suffer from thermal runaway when the gate voltage is inadequate).

So, where is that graph? It's not there as it should be because it only shows the graph when operating at 25C and, that is a significantly bad sign for using this device at tepid gate voltages. You would always use this device at at least 8 volts because there is nothing in the data sheet to give you confidence about using it at lower gate voltages.

For the original part (SPD04N50C3), its front page doesn't say much about it's intended target use so, potentially no problems here because, virtually all problematic MOSFETs state that they are intended for switching regulator applications. Not saying anything at least partially excludes the original MOSFET from having much of a problem but, does it have a graph of ID against VGS? Yes, and here it is: -



This graph speaks volumes about how it will perform with less than adequate gate voltages. Look at the 150C graph and the 25C graph and note the gate voltage where they cross. It's about 5.4 volts. This is called the zero temperature coefficient point because, if you apply that gate voltage and the MOSFET warms up, it will neither take more drain current nor take less drain current i.e. self-heating doesn't change the drain current. No thermal runaway!

If the gate voltage were (say) 4 volts then self heating would start to increase temperature and the device would take more drain current and might destroy itself. That destruction can take place in a fraction of a milli second and the device itself may not even register as being slightly warm. (Ref the Spirito effect).

If the gate voltage were greater than 5.4 volts (say 6 volts) the drain current would fall as the device warms up i.e. you avoid thermal runaway. In my experiece, the graph shown is pretty good compared to most regular MOSFETs and it is for this reason alone, I would not recommend the STD7NM60N for your application.