What's a neutral and popular patriotic term for one's country?
There isn't anything wrong with native land. (It hasn't been tarnished by association with Germany, Russia, or the American Department of Homeland Security.)
And it is a phrase that has been used for centuries in English. Maybe not as often as the other three phrases, but it's definitely out there "in the wild". For an example from literature, Lord Byron wrote a poem Adieu, Adieu my Native Shore around 1815, which contains the phrase my native land. It is taken from Childe Harold's Pilgrimage, which is in part autobiographical, so the phrase here refers to England. And in Thomas Hardy's book The Return of the Native, the title character is a native of Egdon Heath, in England, who has returned from Paris.
I suppose if you were using it with respect to a country like Australia or the United States, where white settlers displaced the original natives, some people might think it was insensitive. But otherwise, it should be fine.
It's true that to a certain extent these words are associated in English with Nazi Germany and the Soviet Union, but these are by no means the only associations, nor should they be.
First of all, I'd just like to point out that one of the major reasons why these words have these associations is not so much because they have been used predominantly by the previous two regimes I've mentioned, but because people in certain parts of the world are exposed to their popular culture rather than being acquainted with facts or knowing about other parts of the world.
To start off, if "homeland" has any mental association with America's Department of Homeland Security (which it may, especially among Americans), it's only possible that this association has existed since 2002, as it was established directly after the September 11 attacks.
The term "fatherland" or something close to it is ancient. In Ancient Greek "patris" can be translated as "fatherland". The Latin word is "patria".
As to "fatherland" acquiring an association of Third Reich Germany, Wikipedia notes that this association began when countries not using the word "fatherland" became acquainted with German propaganda of the era. We can see the meaning of "fatherland" without Nazistic allusions from the current national anthem of Germany, which despite going through some drastic measures of denazification, still maintains in their anthem the line "For the German fatherland!". Nigeria's anthem contains "Nigeria's call obey to serve our fatherland," and was adopted 1978. East Timor's national anthem adopted in 2002 is named "Fatherland". Some countries' official motto include the word (English translation) "fatherland", including Portugal, Cameroon, Dominican Republic, Latvia and Liechtenstein. And here is a list of populations who refer to their country as the fatherland.
Groups that refer to their native country as "fatherland"
As to "motherland", here's how the dictionaries define it:
1.One's native land.
2.The land of one's ancestors.
American Heritage Dictionary2.another word for fatherland
Collins Dictionary
Random House Kernernman Webster's Dictionary defines it as "mother country". Here is a list of definitions of "mother country", in case you want to consider it. "mother country" makes sense to me, in same way your mother tongue is the language you're born into.
Mauritius adopted its national anthem titled "Motherland" in 1968. Monserrat's national song "Motherland" was adopted in 2014. Montserrat is a British Overseas Territory in the Caribbean.
Also, about your point that "native land" sounds like land belonging to aborigines, the "native" is supposed to mean "of one's birth". At the bottom I've shown three national anthems containing "native land" in untranslated English, meaning just this, land of one's birth.
The fact that "fatherland" and "motherland" seem to have been hijacked or perverted through two particular regimes should not cause you to worry about appearing to endorse either's politics. If people were more accustomed to other cultures and words they would be more immune from having these words elicit negative emotions, and seeing them as mainly products of Nazism and Soviet communism, which most definitely they are not.
I personally think it's appropriate to reappropriate these terms for what they really mean. The only way to do justice to your country's patriotic songs are to do an accurate translation. The only way to maintain these two particular words "fatherland" and "motherland" without them continuing to be tainted with these two political ideologies is to rehabilitate them.
If you're worried about appearing to support a particular politics just by using these words, I would strongly suggest, given the above information of the way these terms are used in non-pejorative ways, that you not hesitate in doing an accurate translation. If, on the other hand you wish to do it out of respect for those who are particularly sensitive to these particular words, then that's another matter. Also, I would have your same suspicion that the people posting these comments are trolls.
Some examples from countries' national anthems (All have English as either de facto or de jure official language:
United States: "Land of the free, home of the brave".
Australia: "Our home is girt by sea.... our land abounds in nature's gifts...
Canada: Our home and native land! (Note the French version literally translates to "land of our ancestors/forefathers/forebears")
Antigua and Barbuda: To safeguard our native land
Sierra Leone: Singing thy praise, O native land.
New Zealand: God defend our free land.