No, that's not what the genitive case means. It describes the nature of something, not it's location. There other ways the Greek language uses to locate something.
You clearly don't understand the genitive case. To you it means what you want it to mean, specifically whatever it takes to align with your preconceived ideas. That's just not how any language works, let alone Greek which is arguably the most precise language in all history.
The description of the heavenly these faithful saints looked for is "better", not the same as that which they came out, they could have returned to, but confessing to be strangers and pilgrims on the earth, they seek a country far better. A heavenly home, looking for a city whose builder and maker is God. "Wherefore God is not ashamed to be called their God: for He hath prepared for them a city."
Hebrews 11:10 (KJV) For he looked for a city which hath foundations, whose builder and maker is God.
Hebrews 11:13-16 (KJV) These all died in faith, not having received the promises, but having seen them afar off, and were persuaded of them, and embraced them, and confessed that they were strangers and pilgrims on the earth. For they that say such things declare plainly that they seek a country. And truly, if they had been mindful of that country from whence they came out, they might have had opportunity to have returned. But now they desire a better country, that is, an heavenly: wherefore God is not ashamed to be called their God: for he hath prepared for them a city.
It's clear for those who read this passage without trying to force their doctrines into the passage. These faithful saints of Old longed for a spiritual city, not land on this earth but something far better. Greater describing what they looked for that would be far better than the country and they came out of. They had no desire to return to that physical country and city. They died longing for spiritual fulfillment of Mount Sion, the city of the living God, the heavenly Jerusalem, and to an innumerable company of angels. They had to wait for the advent of the promised Messiah to come to make atonement for sin, and defeat death before they could spiritually ascend to heaven with Him.
Hebrews 12:22-24 (KJV) But ye are come unto mount Sion, and unto the city of the living God, the heavenly Jerusalem, and to an innumerable company of angels, To the general assembly and church of the firstborn, which are written in heaven, and to God the Judge of all, and to the spirits of just men made perfect, And to Jesus the mediator of the new covenant, and to the blood of sprinkling, that speaketh better things than that of Abel.
Ephesians 4:8-10 (KJV) Wherefore he saith, When he ascended up on high, he led captivity captive, and gave gifts unto men. (Now that he ascended, what is it but that he also descended first into the lower parts of the earth? He that descended is the same also that ascended up far above all heavens, that he might fill all things.)