Okay let me rephrase the question without all the commentary:
How were altars allowed in Israel other than in Jerusalem?
The answer is simple enough. The law was: "until you pick the one special final place, there can be other altars. Once you get that special place, all sacrifice will be there." And Jerusalem was that place.
Deuteronomy Chapter 12: (I'm using Aryeh Kaplan's translation, which I highly recommend as he gives context -- unlike the JPS you're using which has been leaving you somewhat confused.)
12:5 This you may do only on the site that God your Lord will choose from among all your tribes, as a place established in His name. It is there that you shall go to seek His presence.
12:6 That shall be the place to which you must bring your burnt offerings and eaten sacrifices, your [special] tithes, your hand-delivered elevated gifts, your general and specific pledges, and the first-born of your cattle and flocks.
12:7 You and your families shall eat there before God your Lord, and you shall rejoice in all your endeavors, through which God your Lord shall bless you.
12:8 You will then not be able to do everything that we are now doing, [where each] person does what is right in his eyes.
12:9 Now you have not yet come to the resting place and hereditary land that God your Lord is giving you.
12:10 But you shall soon cross the Jordan and live in the land that God your Lord is allotting you. When He has granted you safety from all your enemies around you, and you live in security,
12:11 there will be a site that God will choose as the place dedicated to His name.
It is there that you will have to bring all that I am prescribing to you as your burnt offerings, eaten sacrifices, [special] tithes, hand-delivered elevated gifts, and the choice general pledges that you may pledge to God.
12:12 You shall rejoice before God your Lord along with your sons, your daughters, your male and female slaves, and the Levites from your settlements, who have no hereditary portion with you.
12:13 Be careful not to offer your burnt offerings in any place that you may see fit.
12:14 It must be done only in the place that God shall choose in [the territory of] one of your
tribes. Only there shall you sacrifice burnt offerings, and only there shall you prepare all [the offerings] that I am prescribing to you.