No, of course not. But, that doesn’t mean that there wasn’t a similar point that Jesus made …
I would assume that, with Romans (and others) hanging criminals on crosses, the term “take up your cross” (if actually used in those day) would refer more to a consequence of criminal behavior; not a “carry your burden” type connotation.
First of all, let’s consider the context: Jesus has just told His disciples that he will be rejected by the Jewish leadership and killed. There is only one way to kill a person under the law in this context: capital punishment under the Romans. Jesus’ words here (to His disciples and the crowd) must have stung in the ears of the disciples: He just told them he was going to be killed, and now He makes an allusion to that very thing! Where the master leads, his disciples follow – and if the disciples have to bear a cross, what does that say about what the Master will do? We lose a bit of the sense of it in translation: when Jesus tells them about coming ‘after’ Him, He uses the word ὀπίσω, which means ‘behind’. He’s literally talking about them following in His footsteps!
Secondly, we need to look at the grammar here, and the sequence that the sentence presents. The verbs here (‘deny’, ‘bear’, and ‘follow’) are all imperatives: Jesus isn’t presenting a hypothetical, here; He’s literally saying “deny! bear! follow!”. Moreover, both ‘deny’ and ‘bear’ are in the aorist tense, while ‘follow’ is in the present. With the imperative, the force of the expression is that one is to deny and bear – as single events – and then follow (as an action that is ongoing). These two are ‘preconditions’ of sorts: one cannot commit to the ongoing process of ‘following’ without first denying and bearing.
Third, let’s look at the words themselves: the word translated as ‘deny’ is* ἀπαρνησάσθω*, which is a rather forceful word. It means ‘disown’ or ‘strongly reject.’ Here, Jesus is making a bold statement: one cannot be a disciple and care for his own life; one can only be a disciple by placing discipleship as one’s primary goal, at the (potential) expense of all others. The word translated as ‘take up’ (ἀράτω) can mean ‘pick up’, but it can also mean ‘bear’. It really does fit into this string of imperatives: Jesus realizes that the act of disowning one’s own life is a harsh command. In fact, it’s no less harsh than asking a person to share in act of his own execution. In this light, I don’t think that your original objection holds up: the command isn’t focusing on criminal punishment so much as it’s directing the crowd that they must actively cooperate in putting their very lives in second place to their discipleship.