The transliteration into English doesn't seem right to me.
Is this indeed correct?

All the internet articles about Mussorgsky give Modest as his first name. Why do you think it's incorrect?
An early 20th century member of my French husband's family was called Modeste, which in English would be Modest.
It's the standard transliteration.
A letter-by-letter transliteration (which few do) of Мусоргский сould yield variations in which the English double s+ is replaced with the single +s+ found in the Russian, or the ending of the surname becomes +iy+ or +ii depending on which school of transliteration you follow. Google yields a number of hits for "Musorgskiy," "Mussorgskiy" and other variations. However, it's better to stick with Mussorgsky, as this is by far the most commonly used.
His first name, Модест, is always transliterated "Modest"
Does the OP think that the Christian name is pronounced like the English word modest? It's not; the stress is on the second syllable: mo-DEST.
The placement of stress can be a problem. My grandmother's Christian name was Мария, which is transliterated as Maria. She said that when she first came to the United States she sometimes failed to answer when called because her name was being pronounced with stress on the i, as María, but in Russian the name is stressed on the first syllable, as Mária. She solved the problem by anglicizing her name to Mary.
Edited by NorthAmerican.

Mussorsgky
You have an S and the G the wrong way around, but I doubt this was the point you had in mind.
The double S is an affectation of earlier times that has become established, we would write Musorgsky if he turned up today. Likewise we would write Chaikovsky and Rakhmaninov if we didn't already know them as Tchaikovsky and Rachmaninoff. The ий issue has already been discussed.
His first name, Модест, is always transliterated "Modest"
For him, but maybe not for others of that name. In the translated Russian novel I'm reading just now, the transliterations they use would write this as Modyest. I've read a book where they would write it as Madyest, however perhaps that is going beyond transliteration.
You have an S and the G the wrong way around
Good catch! Amazing how the eye auto-corrects.
For him, but maybe not for others of that name. In the translated Russian novel I'm reading just now, the transliterations they use would write this as Modyest.
I suppose there's always a danger when you write "always." Yes, Модест could be written Modyest to show stress and proper palatalization of the 'd', and a Google search shows that there are a handful of academics who have gone this route in regard to Mussorgsky. (Are there any other notable Russians named Modest? None that come readily to mind, at any rate.)