First, patio (and garage) drains commonly plug up, because they get runoff with a lot of sand and silt in it but very low water flows. Dirt coming up ( a cupful or two) while flushing out such a drain is not necessarily an indication of a broken pipe. Even if it was soil and not runoff dirt they saw, it could just be a drain riser broken due to frost heave or patio settlement - you would need to do a camera run to see where and what the prolem really is - usually about $300-400 for a routing (and go with full-diameter cutter routing to clear roots and grease buildup, not just a snake) of the line full-length to the street, plus a camera run to the street - they usually will not run camera unless it has been recently routed out, to minimize risk of getting camera stuck in the pipe. If only an unseated or broken riser in patio area, might only be a several hundred $ job, not thousands unnecessarily.
From here on, assuming you actually do have a broken sewer pipe downstream:
This is definitely a case where you need 3 or more bids, and not just for the solution they came up with - ask for solutions first, then get bids from 3 on the solution you think is best. If you are totally stymied at which plan to go with, get all 3 and then ask a civil engineer who routinely does sewer design to come out and give his opinion on which is best. My personal philosophy is to first avoid low slope runs, then avoid as many turn fittings (as opposed to gradual sweeping curves) as possible, then go for the shortest run feasible.
I presume the line was camera run to see if the problem was one local broken place (usually $500-1500 repair cost depending on depth of line) or due to general pipe deterioration indicating a need to replace the entire line. OF course, type of line and age comes into the decision too, as does the amount of tree root infiltration you have.
Obviously, replacing the line under slabs is costly and avoided as much as feasible (though it should not be ruled out to avoid nasty bends or flat runs), depending on your ground topography and distance to street, as you also want to avoid unnecessary low-slope runs or turns in the sewer line.
$12500 sounds like a tremendous amount, though not unheard of - usual complete replacement runs more in the $3-6,000 range, including slab repair if needed, and it sounds like you might only be talking about the kitchen drain being rerouted, not your entire house sewer line.