You don’t just need self-financing in a risk-neutral world but it’s a much more fundamental principle. If you look at a portfolio that is not self-financing, i.e. you can inject or withdrawal funds at any time, you can hedge any derivative easily. If you can always add the amount of money you need, then hedging becomes trivial. Thus, you use self-financing in the definitions of arbitrage portfolios and hedging strategies. These concepts are even more fundamental then going over to a risk-neutral world.
Look at an arbitrage: a self-financing portfolio which has zero cost but a positive probability of paying a positive payoff. In any state, it pays at least zero. If you drop the assumption of self-financing, you can think of a portfolio that buys and sells nothing (zero initial cost) and hence pays nothing but you inject 1 unit of the currency (numeraire) in the next period. Then, you always have a positive payoff but it does not really capture the idea of arbitrage and a free lunch.
Thus, self-financing is the key property of admissible portfolios.