Professional opinion (I work with leather for a living.)
Leather Milk is highly regarded, but it’s more suited for some things than others (wax/oil finished things, like Horween’s signature line). I find it too difficult to come across for everyday use, and for the beating that armor takes, there’s better options.
Don’t use mink oil. Yeah, in liquid (read: mixed with other fats) it goes rancid in the bottle if mistreated (allowed to vastly change temperature, extreme age). The main issue is that it eats synthetic thread, so unless you know for sure what you have is stitched with linen, hemp, or cotton, using something else will help extend the life of your pieces. It’s main use is on footwear that gets exposed to salt, it helps the leather resist staining. Over time it will leave a haze on your leather that needs to be cleaned up; this isn’t a defect in the oil, it’s just the nature of the beast.
Neatsfoot oil (pure or a mix with mineral or other types) is a tack and saddle staple. For hard formed or molded things, I don’t recommend it unless you’re doing a very light layer as a rescue method because it’s been untreated for a long period. If you saturate your wet-molded item with oil it’ll soften up and lose shape.
Saddle soap is a great cleaner, but it’s not a conditioner. Once or twice a year, brush off any dirt, use saddle soap according to the instructions, wait for it to dry comepletely, then condition. If there’s mold, use a lemon juice or vinegar mix (do a little googling on the type of leather and the cleaning agent for proportions and techniques.)
If you want some water resistance added in, use something beeswax based. It’ll condition and make water bead off the surface a bit more, Fiebings Waterproof Paste and Aussie are both really good. Dr Jackson’s line has a couple of wax based options, too. Again, too much wax and you might soften up hard molded items.
Otherwise? Look for a cream conditioner, and remember that a few light layers over a week are better than one heavy slathered on layer. Tandy has a few options, Fiebings makes a few things, look for something that has directions like “buff on, wait 10 minutes, buff off” and looks like hair conditioner or hand lotion. Just be wary of anything that claims total water resistance or is a cleaner/conditioner, those have either plastics in them that’s will prevent future conditioning, or soap/detergent that will dry the leather out just as much as it conditions.
If you want to use linseed oil, make sure it’s labelled as “boiled,” and DO NOT LISTEN TO ANYONE WHO USES OLIVE OIL. Most of it is cut with vegetable oil, and if you’ve ever opened a bottle of rancid cooking oil you’ll know exactly how rank that gets. There’s a reason neatsfoot is the standard for people who hard use leather every day on horseback, not canola. If you buy it for cooking, please don’t use it on leather, I implore you. Go in to your local Tandy or other leathercraft supply place, or even a tack and saddle retailer, and get something from them.