Why do other religions exist if Yahweh is the one true God?

short answer

The reason there are many religions in the world is because there are many gods who initially were intermediaries between God and man but they rebelled and took worship for themselves.

longer answer

There are three major reasons for the state of the world today

• Adam’s fall vía the deception of the serpentine being of Eve Genesis 3
• The copulation of the “angels” with the daughters of men that led to the corruption of all flesh with the exception of Noah (and his family) Genesis 6
• The Tower of Babel and the splitting of the nations among the “angels” (the sons of God) Genesis 11

Your question is specifically about the last point the Tower of Babel.

Essentially after the flood, man united against The God of Heaven and in defiance of God wanted to make a name for themselves and shake off the government of God from over their life by giving themselves a place in the heavens among God and the sons of God.

“Then they said, "Come, let us build ourselves a city and a tower with its top in the heavens, and let us make a name for ourselves, lest we be dispersed over the face of the whole earth."” ‭‭Genesis‬ ‭11:4‬ ‭

God indeed understood their intentions and instead decided that if man did not want to be governed by God then He would put intermediaries between man and God but confused their languages to split the power among what would become the nations

“When the Most High gave to the nations their inheritance, when he divided mankind, he fixed the borders of the peoples according to the number of the sons of God.” ‭‭Deuteronomy‬ ‭32:8‬ ‭

Hence therefore after their dispersal each nation group had a new language and an intermediary which in the Hebrew is an elohim.

God decided to make His own nation from scratch through Abraham that’s Genesis 12 and it is for this reason that Israel was said to be the smallest of the nations because it was also the youngest

“But the Lord 's portion is his people, Jacob his allotted heritage.” ‭‭Deuteronomy‬ ‭32:9‬ ‭

Also it’s important to understand that God was still the God over all the nations up until this point and the sons under Him.

“For the Lord your God is God of gods and Lord of lords, the great, the mighty, and the awesome God, who is not partial and takes no bribe.” ‭‭Deuteronomy‬ ‭10:17‬ ‭

It’s interesting that the text uses this language because the gods of the nations corrupted themselves.

“God has taken his place in the divine council; in the midst of the gods he holds judgment: "How long will you judge unjustly and show partiality to the wicked? Selah Give justice to the weak and the fatherless; maintain the right of the afflicted and the destitute. Rescue the weak and the needy; deliver them from the hand of the wicked."” ‭‭Psalms‬ ‭82:1-4‬ ‭

Because they corrupted themselves they lost their eternal status and as such they are doing everything they can to keep man subjugated and delay the inevitable judgment to come that was pronounced upon them.

“I said, "You are gods, sons of the Most High, all of you; nevertheless, like men you shall die, and fall like any prince." Arise, O God, judge the earth; for you shall inherit all the nations!” ‭‭Psalms‬ ‭82:6-8‬ ‭

God will inherit the nations again and God has been battling against these gods who were once in His family as sons of God throughout history

“For I will pass through the land of Egypt that night, and I will strike all the firstborn in the land of Egypt, both man and beast; and on all the gods of Egypt I will execute judgments: I am the Lord.” ‭‭Exodus‬ ‭12:12‬ ‭

We are currently wrestling as Christians against these very gods to win back the inheritance of God. Apostle Paul was fully cognizant of the fact

“Finally, be strong in the Lord and in the strength of his might. Put on the whole armor of God, that you may be able to stand against the schemes of the devil. For we do not wrestle against flesh and blood, but against the rulers, against the authorities, against the cosmic powers over this present darkness, against the spiritual forces of evil in the heavenly places.” ‭‭Ephesians‬ ‭6:10-12‬ ‭

in summary

So the sons of God corrupted themselves, took worship for themselves, became gods in their own right, made the nation allotted to them worship them by subjugating them in fear. Hence there is a common historicity among the religions with obvious nuances.

“For the Lord is a great God, and a great King above all gods.” ‭‭Psalms‬ ‭95:3‬ ‭

And NT

“that he worked in Christ when he raised him from the dead and seated him at his right hand in the heavenly places, far above all rule and authority and power and dominion, and above every name that is named, not only in this age but also in the one to come. And he put all things under his feet and gave him as head over all things to the church, which is his body, the fullness of him who fills all in all.” ‭‭Ephesians‬ ‭1:20-23‬ ‭

If this was too condensed let me know so I can expand on it further

Also the English translations were taken from the DSS and the LXX not the MT

The premise in the question :

Does this signify that there is in fact no reason for God to mind the existence of other fictional Gods, which further implies that God actually doesn't care about us and that we are just created and left to our own devices?

. . is incorrect.

God has specifically stated :

Thou shalt have no other gods before me [Exodus 20:2 Deuteronomoy 5:6 KJV]

This was addressed to the nation to which he chose to express the Law by the mediation of Moses. And the history of Israel contains many examples (the Assyrian and Babylonian captivities in particular) of God judging Israel when they followed other gods than himself.

I the LORD thy God am a jealous God, visiting the iniquity of the fathers upon the children unto the third and fourth generation of them that hate me; [Exodus 20:5 KJV]

God does not 'allow the continued propagation and existence of other religions' as can be seen in history and as can be seen contemporaneously by the natural judgments that fall upon idolatrous nations.