@The Last Stand
Makarov rose to power in the 80s, after saving Zakhaev's life. By the time COD4 starts, he already has his place as an independent element within Ultranationalists, with his own men and small private army. You're right, in that once the party takes over officially at the end of CoD4, and especially after the Airport Massacre most if not all Russian troops that invade Western Europe are under his command, or at the very least have his commanders in charge. Think about it, why invade Europe when it was the CIA that was supposedly shooting up Russian Civilians? Russians were out for blood, and Makarov was there to direct their anger in a way that benefited him.
As for Al-Asad, his role is relatively minor: His whole job was to stage a coup to attract as many US troops as possible. Remember the nuke found in Crew Expandable? Asad had a similar, if not the same exact nuke delivered to him by Makarov, and was tasked with luring US troops to the capital and detonating it, to kill as many men as possible while he lived out the rest of his life somewhere in Bahamas...well, that was the plan anyways, before SAS got ahold of him.
The purpose of the nuke was twofold: To benefit both Makarov and Shepherd, who even then were working together. Shepherd got his war, and Makarov got his CIA patsy Shepherd promised down the line. All of this lead to WW3 once UltraNationalists won the war and US accelerated their invasion of the Middle East, after the Airport Massacre: Shepherd wanted to bring back the days of good old armed conflict, to make American Military patriotic and great again, where as Makarov wanted to take over all of Europe...or glass it, should that fail, and bring back the days of Russian Tsars and ultimate supremacy. Of course, each saw the other as a loose end, so before any of that could happen, both went at each other's throats when WW3 has just barely begun, add to that TF141 and all the other factions you meet along the way and you got a plot of a great action thriller. Sadly, the new trilogy failed to capture that magic for me, it was too tryhard and ended up not showing a goddamn thing that mattered, it was just a set of loosely connected setpieces and nostalgia pandering(and unironic glownigger propaganda).
I might be wrong with some of the details here, I haven't played the games in forever but I do believe that's how all this fits together. Of course, things get even more complicated if you believe Black Ops takes place in the same universe, which is actually canon as of the nu-universe with Cold War being a prequel to MW4/MW2019(as it has Zakhaev)...that would also make every single event in the Zombies universe canon in the MW storyline and that's a convoluted chronology that makes Legend of Zelda lore look straightforward and easy to understand.
Anyways, I hope I answered the question, it's rare I get to sperg out about COD these days(usually at the state of it rather than it's previous content)