It has a timing belt. On the surface it’s pretty coincidental but there’s a lot of unknowns from where I sit. Some of those are:
Compression test before the head gasket repair and for the valve diagnosis along with the readings.
Whether the head gasket replacement was due to weepage or severe overheating and if overheating how long did it go on.
Whether valve lash was ever checked and adjusted if necessary.
How they determined the valves were good when the original work was performed.
Whether the timing belt was aligned properly as mentioned by mountainbike.
I could see this thing going in a number of directions so it’s difficult to say for sure what the cause is.
From the get-go, a compression test should be performed on all cylinders in the event of a head gasket fault. This would detect any piston ring or valve problem on cylinders not affected by a head gasket breach.
I don’t know whether they inspected valve lash or not (likely not) but that’s something that should be done every 30k miles.
My personal opinion is that valve lash should be inspected and adjusted with every head gasket job.
Just my opinion, but if I were doing the job on a 15 year old Subaru with 118k miles and especially with a head gasket fault I would do a complete valve job and surface the heads at that time while replacing the valve seals. If it was a warranty procedure or a low miles engine then the valve job might be given a pass but only in that situation.
It’s also unknown how the valve job labor cost is figured. If the amount quoted (250-300) is for grinding the valves and seats only that seems high to me unless they do their own valve work (not likely), the auto machine is high, or they’ve adding a substantial markup to the auto machine shop charge.
Around here it’s 5 dollars per hole, meaning each valve/valve seat. That means even on a 16 valve engine it’s only 80 dollars for the valve job. Surfacing heads, if needed, are 35 each. It could be the area where you live is priced much higher than than where I live.
That’s pretty lengthy and I hope something useful can be gleaned from it. I apologize for not being more precise.