I don't want to be a Debbie Downer here, but I have worked in the restoration business. Nothing Servpro has said is wrong (aside from their timeline, which is a hair excessive, but not ridiculously so. They can't start work without an approved claim due to the way they run their business).

Under a homeowner's insurance policy you're entitled to "like kind and quality", meaning, if your floor goes under your cabinets, you're entitled to flooring that goes under your cabinets. If you have no transition strip between your kitchen floor and living room floor, you're entitled to have the living room floor replaced too. Basically, the work to repair your insurance claim can't devalue your house.

Doing a quick-and-easy flooring job is certainly possible, but if the subfloor starts rotting or delaminating and the cabinets shift, the story on the 6 o'clock news is just as likely to be "scumbag flooring contractor cons expectant couple with shoddy kitchen patch job." There's a right and wrong way to repair this damage - you can pan Servpro for being a little slow, but anyone who can fix this in less than 2 weeks will either be more expensive than insurance is willing to pay, or is going to do a lousy job that hurts the value of the house. There's no third option. This is a month of work plus the claim approval time to do it the right way.

I tend to agree with this, OP seems very patient. This is newsworthy, give them a call and watch Home Depot get a company to make this priority.

Hopefully worst case is a couple weeks in a hotel that is paid for.