"so, is it possible to go AG without mash tun and false bottom if you're still saving up for the right equipment?"
Well, there really is no "right" equipment and no "proper" way. If you've got a couple grand to drop on brew stands and pumps, more power to ya. If not, there is absolutely nothing wrong with mashing in a cooler or BIAB. The grain really doesn't know how much you spent on your mash tun anyway.
Your "sparge" technique was fine and will likely result in a tasty beer. You probably didn't great efficiency, but that isn't a critical with a partial mash since a lot of the sugar is coming from extract still.
You can certainly go all-grain with any of the methods listed above. In homebrewing, high efficiency is more of a bragging right than a necessity. It's more important to be consistent with your mash and then just plan the recipe accordingly. eg 70% vs 80% just means you need to buy $2 more of base malt. No biggy.