A few thoughts
Mortgage companies want a clean credit report. You have no chance of getting approved with unresolved negative items.
Derogatory items do fall of after 7.5 years from first delinquency, paid or not. Paying or settling derogatory items doesn’t restart that clock. It has nothing to do with your liability. It’s only the reporting period on your report.
Collection agencies tend to sue just before the SOL expires and try to get a judgment. Since so many time has passed, I believe they didn’t file a lawsuit against you, or it would have already showed on your file.
Regarding a pay for delete settlement – that depends whether the original charge off is a one time (like unpaid doctor’s bill) or rather an on-going payment such as credit card. Collection agencies can only delete their entry, but the original entry (in case of a defaulted credit card for example) would still remain on your report.
At this point you should be able to settle for less than 10% in a lump sum payment. That would change the status to “Paid” or “Settled”, which should be good enough for the mortgage officer. For a pay-for-delete you will be asked to pay much more.
Just make sure you get any settlement agreement in writing before you pay and don’t give them direct access to your bank account.
Another option is simply to wait out the 17 months, but then you won’t be able to buy the house now.