Russian has different transcription systems for different languages.
Some of them (Japanese, Korean, Chinese) have been developed by a single person or by a group of scholars, who had invested some thought and consideration into them, and had them codified.
Some of them (Greek, Latin, English) have been in use for quite a long time, so the rules have just evolved. Those rules are partly codified, partly implied and partly don't really exist and are just dealt with on ad-hoc basis.
Finally, lots of languages just don't have any kind of tradition of transcription into Russian.
Those systems might be old or relatively new, they might more or less closely reflect the actual pronunciation of the original language, they might have evolved along with the original language and/or Russian language, and they might be ridden with inconsistencies.
They also, usually, are not consistent between each other.
Now, an ideal transcription system has several requirements.
Russian renditions of foreign names should resemble the original pronunciation; should resemble the original spelling (if applicable); should be reversible (ideally you should be able to reconstruct the original word just from looking at its Russian rendition); should be actually utterable by Russian speakers; should be unequivocal and so on.
Of course all those requirements are impossible to achieve at once (partly because the original languages and writing systems are not examples of order and logic either), so some compromises are being made, either by the authors of the system or by the people who learn it by heart from their predecessors.
All these compromises have their ups and downs, pros and cons, and different people have different preferences about what should be put on top priority when making these compromises.
That's why some systems gain traction and some don't: if you don't make enough people happy enough with your system to start using that over the others, yours would just die from natural causes and be consigned into oblivion.
That said,
This apparent inconsistency is appalling to me
I'm sorry that you feel this way. This is something that is going to happen often.
If you say that non-Russian names should be blindly transliterated to Russian
Non-Russian names is a very large group of names and there's no single "you" who would have a say about all of them.
They come from different languages, get transcribed by different systems designed and used by different people.
English and French transcription rules have been never designed, they have evolved over the centuries.
People used to write and say Робинзонъ Крузоэ, сочиненiе г-на Данiеля Дефоэ but at some point that had been considered stupid so everyone stopped doing that.
People also used to write and say комета Галлея and круги кровообращения Гарвея and still do, because this is considered OK.
Inconsistent? Yes. Hard to learn? Yes. That's how evolution and collective decisions work.
If you say that it is because Erdoğan is the name of a president and not a geographic name like Salisbury, then what about Ronald Reagan?
The thing about Ronald Reagan is that he had an English name, and President Erdoğan has a Turkish name, so see above.
Personal names, or the status of their bearers, as a matter of fact, are rarely if ever get special treatment in transcription systems. There might be traditional exceptions, like Авраам Линкольн, but it's not a part of a system. President Lincoln's name is an exception because he was born and became famous enough to get an established Russian transcription back in XIX century, and not just because he was elected president.
It is hard to see why the Russians cannot do what the Albanians can
Just the fact something is good does not mean it's going to be adopted.
We'll leave aside the question of whether or not Albanian system is that great for a moment.
To "do" something with the language, like introducing a new transcription system, you have to gather a critical mass of influential people who can make that happen, leading by example.
Albanians don't make such a critical mass for Russian speakers.
Russians and Albanians live in different countries, speak different languages, write in different writing systems and don't share literary traditions in any significant way (not that I'm aware of).
There were never a critical mass of people who would be 1) influential Russian grammarians 2) familiar with Albanian language, 3) fascinated with the Turkish to Albanian rules and 4) not happy with the Turkish to Russian rules to make this transition happen.
Now, to your question:
Why is the surname Erdoğan spelt in Russian as Эрдоган despite that ğ has no sound in the Turkish language?
I don't speak Turkish and can't even say if your premise is true, as some sources say it's still pronounced in some dialects. But let's assume it's true.
At some point, a compromise had been made, most probably implicitly, between several conflicting aspects of a transcription system.
The Turks do one thing in writing (put the ğ in their words) and another one in speech (leave a hiatus between the vowels).
A hiatus between the vowels is not the same as two vowels uttered together, and this may change the meaning of words as far as I understand, so this is undoubtedly a phoneme.
To mark this phoneme in writing, they used a letter which was not a part of standard Latin alphabet.
I'm not sure what did they use when they were writing in Arabic script, but for some reason when switching to Latin script they chose to base this letter on the letter g, and not, say, x or p. They must have felt that the G was a better choice.
Russian transcription system had to face the same choice. Russian could have just omitted the letter, could have used a diacritic, could have used another letter than г or could have used г.
Some Russian author, probably long ago, had made a decision that it should be a г.
It has some upsides: it somewhat resembles the pronunciation at least in some dialects, it somewhat resembles the writing, it's some kind of a clearly marked phoneme.
It has the downsides either: it can be confused with the g, it is not actually pronounced in the mainstream dialect and it makes Mitsuko sad.
In other words, it's a compromise.
Is it a good compromise? I don't really know, it's not a large part of my life and I've never given any thought to that before you asked this question.
Apparently, it was good enough for everyone concerned to gain some traction, as that's what Russian ended up using.
I am also curious whether it will be really considered a mistake if I write "Эрдоан" (i.e., without г) in a Russian text of mine.
Usually, if you are writing or saying something not the way the others do, people would start looking for reasons behind that. Why does Mitsuko write Эрдоан and not Эрдоган like everyone else? What does she want to say with that?
And people have only so much attention and interest in reading, especially these days, so if questions like this would interfere with the rest of your text and stay unanswered, the chances are the readers would lose interest more quickly.
Using an alternative spelling is a literary device, a way to draw attention to some aspect of your prose by leaving a thorn your reader's attention would catch on.
If that's the effect you're hoping to achieve with the non-traditional spelling, then by all means go for it.
If that's not something you're looking for when writing your text, then concentrate on your main message and leave the less important stuff (like Turkish names) the way people are familiar with.
