What matters here is the chemistry.
Modern anti-perspirants rely on wrapping the active aluminium salts in a fatty-acid ( like a wax) coating. This wax gradually breaks down during the day, releasing new active salts all the time.. It's how they claim "24hr protection" etc. These fatty-acid residues are what stiffen your shirts.
To get rid of the hard pit-crust, you need something chemically stronger than the stuff you're trying to remove.
For a start, detergents won't work. The aluminium salts strongly repel anything water bearing. Oxy.. cleaners are not reactive with pit cake either - it's like trying to remove oil with water.
Vinegar is just about acidic enough to react with the waxy salts, and may be OK if done regularly on light stains.
But honestly, when that 1977 Led Zep tour Tee is headed for the bin, you need to go up a step...
Sulfamic acid (NOT sulphuric..). This is found in some coffee machine descalers (e.g. by Bravilor), also the main component in Fernox DS3 central heating cleaner granules. Dilute, spray on the crusty pits, leave to react, wash as normal. Works for me.
Hydrochloric acid. The active ingredient in the Deo-Go product from John Lewis. Available in Cement Stain Remover from builders' merchant. Same for Phosphoric acid, same as above.
These are dilute acids, not particularly hazardous, and won't give off dangerous fumes. Certainly no more dodgy than oven cleaners, so take the same precautions (gloves, glasses).