The Roman Empire--and the Republic--did have patrilineal succession. The "fluke"--indicative of the high mortality/infertility rate--is that in the first 200 years of the Empire, there were only two sons (brothers) available when the emperor died. For example, the pre-Empire Julius Caesar died childless in 44 BCE, only adopting his grand-nephew Octavian-turned-"Caesar Augustus" in death.
Augustus's only biological child was Julia the Elder, whose two sons Gaius and Lucius, died of illness before they could succeed their grandfather. Hence, Tiberius was adopted by Augustus and became emperor in 14 CE.
Caligula was "only" the great-grandson of Augustus through the maternal line, but was adopted by his grand-uncle Tiberius after Tiberius' only son Drusus Caesar's death in 23 CE, and was succeeded by his uncle Claudius in 54. Claudius adopted the son of his fourth wife, because his own sons died at an early age--one possibly poisoned by Nero/Nero's mom as a threat.
Nero was the last of his dynasty, his daughter having died at a young age and he himself killing himself at the age of 30 in 68 CE.
Vespasian "won" the civil war in the Year of Four Emperors, and he was actually succeeded by his son Titus in 79 CE, and then his other son Domitian, who was assassinated. Nerva succeeded him based solely on his service in the imperial household and the support of the Senate, but under pressure, he adopted the popular general Trajan.
Trajan adopted Hadrian, Hadrian adopted Antoninus, and Antoninus adopted Marcus Aurelius, whose natural son Commodus "ended" the Pax Romana.
Long story short, this isn't to say that blood had nothing to do with anything in the Roman imperial world--as shown above, if people could "keep it in the family" in Rome, they would--but in light of the low birth rate among the imperial class (I'm not sure about the lower classes), the name--adopted or not--meant more than blood in Rome.
On another note, in many medieval kingdoms--most notably the Franks, who followed Salic Law--the sons shared in the inheritance, and so the kingdom was divided (one of the reasons Charlemagne's empire over time became divided into what we know today as France and Germany.
The Romans did pass down property from father to son, generally speaking. Sure, they had wills for the dispensation of property after their death, but generally speaking things did follow a father to son chain of inheritance. What you might be thinking of is the way in which succession worked in the Empire, but that also went father to son. Tiberius was Augustus's son, though he was adopted. Tiberius's heir was his adopted son Germanicus, then his actual son Drusus, then his adopted son's son Caligula and his son's son Gemellus, but the former was able to remove the latter. Then it passed to Augustus's stepson's son, Claudius (so it went back and traced descent down a different line from the patriarch). And finally it passed to his great-nephew and adopted son, Nero. What mattered to the Julio-Claudians was some form of descent from Augustus, even if via adoption.
Then, after the Year of Four Emperors, we have the Flavian dynasty of Vespasian, his son Titus, and Titus's brother Domitian. Domitian inherited because Titus had no male heirs. Domitian was assassinated and the Senate chose Nerva, who had no children. Nerva adopted Trajan who succeeded him. Trajan had no children and adopted Hadrian. Hadrian had no children and adopted Antoninus Pius. Antoninus Pius had no male heirs and adopted Marcus Aurelius and Lucius Verus. Verus died leaving Marcus Aurelius sole emperor and his son Commodus became emperor. Commodus was assassinated and then we have the Year of Five Emperors.
Let's stop there. So, what do we see? Very few biological sons (and very few children in general). Augustus had no sons, Tiberius had no surviving sons, but a surviving grandson who was supposed to be co-Emperor. Caligula had no sons. Claudius had a surviving son, Britannicus, but he was too young and Claudius had issues with his mother, to put it mildly. And then Nero had no sons. Then Vespasian had sons, but his son Titus didn't and his second son was removed and replaced. Then we have a series of sonless emperors until we get to Marcus Aurelius, whose son did inherit.
The other answer to your question is that the Germanic tribes that formed these new kingdoms that replaced Roman rule had their own rules about how inheritance worked. So even if the Romans had truly not followed a patrilineal model, then the new kingdoms would have anyway because of their own legal traditions that used patrilineal descent. We see this most famously in Salic law, but all of the different peoples had their own way of determining inheritance and descent, which was often (if not always) patrilinear.