What is the relationship between Billy's current Add-ons and the new add ons in PTB?

Can't confirm any of this. But I tried to compile the most obvious ones:

• Depth Gauge Rake -> Steel Toe Boots
• Light Chassis -> Low Kickback Chains
• Steering Boots -> Dad's Boots
• Chainsaw File / Homemade Muffler -> Off-Brand Motor Oil
• Homemade Muffler -> Apex Muffler (depends on whether the gray Chainsaw File turns into the yellow Off-Brand Motor Oil)
• Carburetor Tuning Guide -> Tuned Carburetor

Those ones make quite a lot of sense, but I'm not too sure about the other ones. I think most of the new "weird" addons (Mother's Helpers, Black Grease, Big Buckle) will originate from the chain addons (Grisly Chain, Rusted Chain, Begrimed Chain) or alternatively Long Guide Bar. Since Carburetor Tuning Guide is most likely to change to Tuned Carburetor, I assume that the old chainsaw cooldown addons (Thompson's Mix, Shop Lubricant, Vegetable Oil) will turn into anti-overheat addons (Junkyard Air Filter, Punctured Muffler, Pighouse Gloves).

I remember when they changed Nurse they turned some gray/yellow addons into iridescent ones (that's why I got 70+ of those even though I certainly don't main her). Therefore I'd only advise you to use up your increased charging addons (if you enjoy instasaw). They are certainly going to be nerfed, if the current PTB doesn't receive a bigger change. However, nothing of this might happen, so you shouldn't rush it.