I will address your question: How can we reconcile 1 Corinthians 7:10-11 with Matthew 18:15-17?
Very good question. The answer lies with understanding the complete context of each section of scripture.
1 Corinthians 7: 10-11 can stand on its own. Paul is reinforcing the fact that the Law of Moses allowed divorce and Jesus did nothing to change that. Remember, Jesus came to fulfill the law and not destroy it plus He also said that not one jot or tittle of the law shall pass until all is fulfilled. Paul in 1 Corinthians 7 is just encouraging the divorced person to be reconciled to the partner.
You very correctly reference Matthew 18 as applying to cases of divorce but you need to remember that Matthew 18 is not a “church discipline” chapter as many believe, it is the reconciliation and forgiveness chapter.
Most people see Matthew 18: 15-17 as “get your brother to see the error of his ways and get him to repent”. If you look at the actual words (KJV) you will see the word used is “hear” and not “repent”.
15 Moreover if thy brother shall trespass against thee, go and tell him his fault between thee and him alone: if he shall hear thee, thou hast gained thy brother.
16 But if he will not hear thee, then take with thee one or two more, that in the mouth of two or three witnesses every word may be established.
17 And if he shall neglect to hear them, tell it unto the church: but if he neglect to hear the church, let him be unto thee as an heathen man and a publican.
The context here has nothing to do with repenting and everything to do with communication and reconciliation. Remember, for all we know the “offended” party may be the one in the wrong. The emphasis here is communication and agreement. The process proceeds from step to step if there is no “hearing” ie, no agreement, no reconciliation.
Understanding verse 17 is critical to understanding the entire section. This verse does NOT say, “if your brother doesn’t repent after the church comes to him, then excommunicate him”. Again, this is not about repenting but about hearing and reconciliation.
Verse 17 is stating that if there is no agreement after the church has been involved then the bond between the two disputing parties is dissolved. That is the intention by the reference to “let him be unto thee as a heathen and publican”. There is no “obligation” between the Christian and the heathen. This is the same principal as 1 Corinthians 7:15:
15 But if the unbelieving depart, let him depart. A brother or a sister is not under bondage in such cases: but God hath called us to peace.
Here the “unbelieving” is the same as the “heathen or publican”; ie there is no bond between the two of them. Again, nothing to do with excommunicating the supposed offender.
Verses 18 – 20 reinforce 15 -17.
18 Verily I say unto you, Whatsoever ye shall bind on earth shall be bound in heaven: and whatsoever ye shall loose on earth shall be loosed in heaven.
19 Again I say unto you, That if two of you shall agree on earth as touching any thing that they shall ask, it shall be done for them of my Father which is in heaven.
20 For where two or three are gathered together in my name, there am I in the midst of them.
In verse 18, the “binding and loosing” is a reference to the opinion of the 2-3 counselors (see verse 16) in helping the 2 disputing parties to reconcile. If the counselors find that these two should remain bound then they should remain bound and if the counselors find that they should be loosed (irreconcilable differences), then they should be loosed from their relationship/contract/bond. In short, versed 18-20 state that whatever the counselors rule as an outcome of the counseling, heaven will abide by their ruling for where 2 or 3 are there attempting to bring reconciliation in the name of the Lord, then God is there in the middle of the process.
I will stop right there for I believe this answers your primary intention to reconcile the apparent disparity between 1 Corinthians 7 and the idea of “let them remain separated” and your belief that Matthew 18 was encouraging excommunication of the offending party.