Not only money order but there are other internal money transfer services offered by different companies, I mean services similar of Moneygram. I hugely doubt anyone will accept cash via mail from random user. Also, if he and recipient live in USA, then Money Order is okey but if they are from different countries, this can turn into a serious problem. There are countries where it's illegal to send money via post, it will be confiscated and you'll get some problems.
Personally, I wouldn't recommend any escrow to take part in such cases (nothing against OP). There can be problems from recipient like money is fake and then sender may say I sent real money and he sent you fake one, etc.
I have never heard that the form of Money Order is illegal any where in the world. What I know is Money order is legal and accepted almost every part of the world and you can even send money orders to other countries too.
In the country where I live Money Order is sent through government's post services. So, it is completely legal and safe (except if the post officer himself steal your money).
This website
https://www.usps.com/international/money-transfers.htm provides services to send money internationally to several countries as you said.
Moneygram and Westrenunion are good options too. Even I prefer Westren Union more than Money Order and Moneygram.
Where I live, it's seriously prohibited to send money inside country via post and it's especially prohibited if you receive banknotes from foreign countries, it gets confiscated and you may not end up well with the law. Illogical, right? There are a lot of illogical laws and situations in some countries are different from USA like black and white. That's why I stated that it's very captious case for escrower from both sides, any side can argue fake and argue that money they received was illegal (in reality it can be true or not). That's why it's better to get rid of 3rd party persons in this case and use Western union and similar companies. Personally, considering the risks, I advice everyone to just get rid of such cases and don't risk to be a middleman.
