If the type of testing you are doing now is an ongoing issue where an engineer, consultant, technician, etc has to extract the wires from these one-time use connectors (or at least they seem to appear to be). Then the costs of that need to be looked at and weighed against the cost of the connectors.
e.g. suppose it takes ~10 minutes to extract the wires from the connector and the person doing this is making $50/hr. That means to remove the wire (which would take a few seconds with a better connector) just cost the company $8.33 to have them remove the wire from the connector. If the production of this board is 10,000 units a month and the failure rate is 2% then you would expect approximately 200 units non functional units. This means 2000 minutes may be required the removal of the wire on all of the units or $16,666 of direct labor cost. If the cost of the connector is $0.10 difference then the production run would cost $1000 more. The connector cost difference would have to be greater than $1.67 to exceed the labor cost of removing the wire.
Now this is a very simplified example (ignoring a lot of overhead costs) that may (or may not) be representative of your situation. But if you can prove to the company that swapping the connector is more than a convenience for you, but saves the company more money then you can justify the switch to a potentially more expensive connector. Of course this is assuming that out of the 200 units that the majority are repairable, and only a small fraction end up in the bone pile that gets scrapped.
I've seen some situations where an inexpensive board is being worked on for days with labor costs in the $1000 on a board that costs <$100. This usually happens when some short sighted decision was made to not build more units than needed (with the expectation of failures) to save "costs", which end up costing the company more money in labor costs (and in other work not getting done) than they "saved" by not having a larger production run. If this board you are working on is in this category then suck it up and just take the money (as a consultant) and spend the extra time extracting the wires, because if they realize you are costing them more money than the board is worth, then you'll be out of a job.