The reason the EU has done nothing about Syria, as you claim, is because it is now clear that nation building does not work. France and Britain intervened in Libya and it is now a mess. They did not intervene in Syria, pre-occupied with Libya, and it became a mess. Then Russia, Iran, Turkey and Saudi intervened and it became an even bigger mess. Other countries cannot sort out Syria. Syrians will need to do that or somehow the whole world must come to an agreement for a solid plan on Syria. But accusing the EU of inaction while Turkish actions have not solved the problem, is hypocritical.
Yes, Turkey is taking on a huge refugee burden. That doesn’t immediately mean the EU has to help. It also does not mean Turkey can expect more than Lebanon or Jordan, who host even more refugees per head of the population.
Money to Turkey was to be paid in installments and not all upfront. And of course Turkey has done plenty of things to ruffle feathers with the EU. Erdogan needs to change his tune to European standards to get what he wants. And all in all, if the EU pays a few hundred more to have peace at the borders, that is a small price to pay. That said, Corona seems to have stiffened resolve in the EU to guard its borders. Soon, Turkey may find refugees simply cannot cross even if Turkey pushes them.
This is all a dirty business. Refugees have become bargaining chips.