First of all, I agree with your understanding that Onesimus is an escaped slave belonging to Philemon. There's support for this in verse 15 (all quotes are NASB):
For perhaps he was for this reason separated from you for a while
If Onesimus had been sent by Philemon, there would be no reason for Paul to propose a reason for the separation. There is further support in verse 17 (accept him) and 18 (if he has wronged you) that clearly indicate the likelihood that Onesimus has offended Philemon in some way.
Paul makes only four explicit requests in this letter:
- "Accept him as you would me" (v. 17)
- "Charge [any offense] to my account" (v. 18)
- "Let me benefit from you" (v. 20)
- "Prepare me a lodging" (v. 22)
The first two requests are for the benefit of Onesimus. As a runaway slave he could have been punished severely, but Paul is asking Philemon to view him instead as a brother in Christ (v. 16) as Philemon would view Paul (v. 17). Paul then insists on being held responsible for wrongs done by Onesimus (v. 18).
The last request is that Philemon prepare a place for Paul to stay in hopes that he will be released from prison and able to visit.
Paul is not as clear in his third request. What sort of benefit is he requesting? I think the answer is in verses 12-14:
12 I have sent him back to you in person, that is, sending my very heart, 13 whom I wished to keep with me, so that on your behalf he might minister to me in my imprisonment for the gospel; 14 but without your consent I did not want to do anything, so that your goodness would not be, in effect, by compulsion but of your own free will.
Paul is asking Philemon to send Onesimus back to minister to Paul on Philemon's behalf. As the owner, Philemon is free to instruct his slaves to perform any work he desires. If he instructs Onesimus to "go minister to Paul", then Onesimus is serving his master Philemon by serving Paul. Philemon is helping Paul by sending Onesimus as his proxy to minister to Paul. Paul gets the benefit of Onesimus's ministry. In short, everyone wins.
...but perhaps implicitly.
Even though I don't see Paul explicitly asking for Onesimus's freedom, he follows up with:
21 Having confidence in your obedience, I write to you, since I know that you will do even more than what I say.
The even more could have been a very thinly veiled request for his freedom especially when combined with verses 15-16:
15 For perhaps he was for this reason separated from you for a while, that you would have him back forever, 16 no longer as a slave, but more than a slave, a beloved brother, especially to me, but how much more to you, both in the flesh and in the Lord.
Given the salvation of Onesimus, Paul is emphasizing that Philemon should view Onesimus primarily as a brother instead of as a slave. Their legal relationship is still that of master and slave, but in reality they are brothers in the Lord and that status should dominate their relationship.
Would it have been possible for Philemon to continue to own Onesimus as a slave yet view him as a brother? Sounds doubtful to modern ears, but I really don't know enough about slavery in that day to say for certain.