Listen up...the second link you provided also makes NO mention of Syrian government forces (SAA), only local militia which operate under the loose banner of a National Defense Force (NDF).
Every Syrian village in government control has a National Defense force, and they’re set up to provide a modicum of organized resistance to roving bands of Western trained jihadists that plague the countryside.
The NDF are just local volunteers trained in civil defense with a couple of machine gun mounted Toyota pickups, and are otherwise not fit for service in the Syrian regular army.
After posing for some ‘fake news’ photos on the outskirts of Afrin, I’ll wager the mighty NDF hightailed it right back to whichever Syrian government controlled villages in the next province they came from.
There were no Syrian government forces fighting in Afrin and no agreement with the Syrian YPG to do so. To suggest otherwise is 100% fake news.
So the Kurds did ask (beg) the Syrian Army to defend the border against the Turks, but that did not work for them - the Syrian Government only made a fig leaf display. Tried - failed.
That is because the Syrian Army can not stop the Turkish Army, and everyone knows it. They still can’t. So asking the Syrian Government to defend them is pointless - they must ask Iran, who runs the ground war for the regime side, for any serious chance. Iran also currently lacks the capability to stop a determined Turkish attack there, but at least they pose a bigger deterrent.
In any event, the decision had already been made by Assad, Iran and Russia to trade the Kurds in Afrin, in exchange for the Turks ordering a bunch of the Sunni jihadis they sponsor to give up the fight in Syrian heartland areas like East Ghouta in Damascus, Hama and Homs. They ran buses for months, moving the Sunni jihadis and their families out of their old neighborhoods, and up into Idlib and Afrin, while the Turks drove out Kurdish populations from Afrin.
There is almost nothing left of the old Syrian Arab Army - even the few units they do field, have their ranks heavily salted with foreigners provided by Iran. In some cases they are almost entirely foreigners with Iranian officers (like around Deir ez Zour), carrying the old guidons, with just a few token Syrian military. The few units that are strongly Syrian or Alawite are kept close to the heartland and Latakia respectively, except for the Tiger Force which provides their limited armor for assaults.
In the many Iranian provided militias, like Hizbollah, there are basically no Syrians. As for the NDF that was sent to Afrin by the Assad Government, “The creation of the NDF was personally overseen by Iranian Quds Force commander Qasem Suleimani.” according to Wikipedia. Few of the ones that went into Afrin spoke Arabic.
The Iranians have little interest in seeing the SDF Kurds preserved. It will serve their long term designs of dominating Syria like they do Lebanon better to have Turkey crush them, so Iran can re-form the remnants into more of their own surrogate militias. Even fewer of the Syrian Kurds are Shi’ite than are Iraqi Kurds - overwhelmingly Sunni, secular or other. They are a natural problem to the Iranian vision for the region.
The strategy of the Kurds just rejoining the Assad Government was long a feasible option, until effectively Iran took over. It may become an feasible option again if sanctions (or regime change) weaken Iranian control enough, or if Assad gets a broad peace deal, and the time and money to rebuild. But Iran will not willingly give Assad time to regain control if they can help it - they will actively organize the population and manipulate politics like they did in Lebanon, to consolidate Iranian control.