Of course ACuriousmind's answer is right, but I would like to add a few things.
I think I understand where your confusion lies, and you have come to a cornerstone in QM. This is why QM is a beautiful thing, it is really different from classical mechanics.
Your confusion lies in that you are trying to go from up to down in scale, you are trying to go from bigger to smaller and use the same methodology. You are asking how a certain elementary particle can convert into another elementary particle, the same way a composite particle, or a macro object would convert into another composite particle or macro object.
This is more of a classical view, and it pre-assumes, that everything is made of the same constituents. True, in strong theory this might work, and this way if you want you can explain that every elementary particle can be converted into any other (except if it is forbidden by laws of physics).
Now in our currently accepted theories, SM and QM, this does not work this way. You cannot say that an electron is converted into a neutrino just because they have the same kind of constituents. Both are elementary particles.
You are saying that chemistry works that way and it is true. Only because in chemistry, everything is built up by atoms, chemistry does not go deeper and you can use this classical view of converting chemical elements into another one.
In QM, everything is energy(matter, particles), and in an interaction it is just simply converted into energy, if you will, into other forms of energy (other matter,particles).
This is why in your interaction, you have to look at the whole interaction, the total energy, and that has to stay the same, this is conservation of energy. This is the basic rule that you have to follow.
Let's see electron capture.
Electron capture (K-electron capture, also K-capture, or L-electron capture, L-capture) is a process in which the proton-rich nucleus of an electrically neutral atom absorbs an inner atomic electron, usually from the K or L electron shell. This process thereby changes a nuclear proton to a neutron and simultaneously causes the emission of an electron neutrino.
There are two types of electron capture:
an outer electron replaces the missing electron (the inner one that was absorbed by the nucleus), and an x-ray is emitted (energy equal to the difference between the two shells)
auger effect, the energy is not emitted in a form of a photon, but the energy knocks off the outer electron
Now you are saying that you were told in the parent question that the electron is not included in the neutron after the capture. This is not true. The nucleus absorbs the inner electron as energy (not in the form of an electron). The inner electron gets transferred into energy, and that energy gets absorbed by the nucleus.
An electron is defined as an elementary particle, not consisting of anything else, it does not have any internal structure.
The proton in the nucleus is made up of a sea of quarks, antiquarks, gluons, antigluons. This is an ever changing sea, and only if you net them out, will you get three valence quarks. Now yes, these quarks happen to have exactly 1/3 of the EM charge of the electron. And yes, the electron is able as per QM to transfer into energy and interact with the proton, its quarks, antiquarks, gluons, antigluons, and thus convert the quarks so that the remaining quarks will have a different net EM charge (in this case neutral). This is how you get a neutron from a proton (one way to get it).
The neutrino that is emitted during this type of beta decay (yes, electron capture is a type of beta decay), is not a converted electron, and it is not the same as the inner electron that was absorbed by the nucleus in the form of energy.
The laws of physics are so that the total energy must be equal before and after the decay, and if the neutrino was not discovered (in beta decay), the total energy did not add up.
After they discovered the neutrino during beta decays, it became clear, that there must be an elementary particle, that has exactly the energy that was missing from after the decay.
The simplest way to think about it is not classical, but QM. The proton absorbs the electron:
all of the EM charge of the electron is absorbed by the proton, that is how it is able to convert into a EM neutral particle, the neutron
not all of the electron's kinetic energy (and rest mass converted into energy) is absorbed by the proton to convert into a neutron, it does not need all of the electron's kinetic energy (and rest mass converted into energy), so there must be an excess of energy
To make the neutron stable, this energy needs to be released, and it is released in the form of a neutrino. It needs to be released in a form of particle, that is:
it's total energy equals the excess energy of the inner electron, that was not needed for the proton to convert into a neutron
So as per QM, the emitted neutrino is not a converted inner electron. The inner electron converts into energy, and part of that energy is released in the form of a neutrino.
The way you are talking about this as chemistry, would suggest that the electron and the neutrino are made up of the same constituents. So far, all experiments fit the theory of SM and QM, showing no internal structure for the electron.
Maybe if string theory proves right, then we will see that the inner electron that is absorbed by the proton, made up of quarks, and the neutrino, will all be made up of the same constituents, then, and only then, will you be able to make this decay look like the classical chemical reactions.
But for now, as per QM, we say that these are all converted into energy, and convert into other forms of matter, particles. The reality is, you are asking how, and we do not know, we do not know how they get converted, we do not really know what really gets converted, and for the theory to work, we call it energy, all matter and particles are convertible to other forms of energy, other types of particles (along the lines of laws of physics).