If a neutron decays to proton + electron, and a proton can decay into neutron + positron, doesn't this mean neutron = neutron + electron + positron?

A neutron does not have the same mass as a proton plus an electron, and a proton does not have the same mass as a neutron plus a positron.

• Proton mass = 938.272 MeV
• Neutron mass = 939.565 MeV
• Electron or positron mass = 0.511 MeV

https://physics.nist.gov/cgi-bin/cuu/Value?mpc2mev|search_for=atomnuc!

So $m_p+m_e = 938.783 \text{ MeV} \ne m_n$ and $m_n + m_e = 940.075 \text{ MeV} \ne m_p$

Now, your unstated question may be based on thinking that because a neutron can decay into a proton and an electron (and an anti-neutrino) that must mean that it contains a proton and an electron (and an anti-neutrino). That is not the case. When a subatomic particle decays into a different particle the new particles are created, they are not merely separated out of the previous particle. The masses may not balance and the difference is the total KE of the products.

Dale's answer is obviously correct but I'd like to add another way to look at this.

When a neutron decays into a proton and an electron (and an anti-neutrino), while it's true that it doesn't mean that a neutron was a bag of a proton and an electron (and an anti-neutrino), mass conservation should still hold. And it does hold. The mass of the combined system of the proton, the electron (and the anti-neutrino) would be exactly the same as the mass of the neutron. This works out because the mass of a system of particles is not the sum of their masses. Relativity tells us that what gets added together is the four momentum of the constituent particles and you get the mass of the combined system by squaring the total four momentum which is not equal to the sum of the masses of the constituent particles unless there is no relative motion between the constituent particles. So this tells you that a neutron, in fact, can't decay to a state where the proton, the electron, and the anti-neutrino are moving with the same velocity.

A neutron has more mass than a proton and an electron, so it can $\beta^-$ decay, and it does.

A proton does not have more mass than a neutron and a positron (which, as you point out, is the same as the mass of an electron). The laws of arithmetic still hold. So a proton on its own cannot $\beta^+$ decay.

But there are some nuclear isotopes in which changing a proton to a neutron produces a new nucleus with greater binding energy. For example, $^{22}$Na has a mass of 21.994 MeV and can decay to $^{22}$Ne and an electron with 0.003 MeV to spare, as the protons and neutrons in the Neon nucleus arrange themselves more tightly than in the original Sodium. It's in cases like these where the total nuclear energy balance is favourable that positron emission can occur.