First off, for your question -- nah, wouldn't worry about the synthrapol. I think formulation of dyes is a bit different than it used to be. Also, is it possible that the collar fabric is different than the rest of the shirt? Sometimes I think 100% cotton gets woven with a bit of something else to make it 102% total.
I did a lot of fabric treatments 20-30 years ago and then moved on to other things. Last year I bought a beautiful bag of hand dyed fabrics at the thrift store. I could feel they had been treated in some way, but it turned out they had never been properly dye set. Went to all the bother of being starched and pressed but not stabilized...
First I machine washed like colors in the washer with my normal detergent. Since I didn't know what had been used/how they had been processed I decided to basically heat set the "stain" (the dye) by boiling for 20-30 minutes in a bath also containing both salt and vinegar, rinsing until clean, then another 20 minutes in the microwave, rinse again... and only then I washed in the washer with a color catcher. Some still required another round, most were fine by that point.
Since then my quilting friends have been handing me problem pieces and I just had to do a similar thing on two other pieces last week, one of which was commercially made the other I don't know but that thing ran so much it went from a very dark to a medium fabric.
I also tried dying some Kona white into more reds for my project. I had terrible results using Rit products they I used to use all the time 20 years ago. The dyes just would not set. The pieces looked so good and all faded to pepto pink.
I strengthened the color by acting like I was starting with fresh fabric. First I washed it again adding caustic soda to the wash. I found you can easily buy it in reasonable quantities these days -- at Walmart! Next to the fabric area they have crafts and had a lot of tie dye t-shirt stuff and have small packets there. There was also a Rit Dye display in with the quilting notions -- but check out the tie dye stuff!
Then I overdyed it. I am getting my best results with boiling on the stove top. 20 years ago I did tie dye t-shirts in the microwave that still look great, again, when I tried that with the red I just got pink. Also was unimpressed trying to dye in the washing machine. I sacrificed my aluminum canning pot (my biggest pot) to being no longer food worthy, stainless should be fine. Glass in the microwave is perfectly fine.
I think the best dying now is being done on rayons... I had signed up for a Ricky Tims thing this spring (has a segment on dying) but it got cancelled due to Covid.