Now that 'we' know where the asBestos is, you have a couple of options.
1) Run. If you don't like the situation in any respect, even if the Seller had it removed, then run.
2) Ask for either Escrow $$$ that covers the ENTIRE cost of the asBestos removal or get that same amount of $$$ as an ADDITIONAL deduction from the selling price. Either way, the Seller covers the fee. If they only offer to pay HALF of that cost, well ..., that's up to you. Maybe the next potential buyer is better off to deal w/it. You can always say all-or-nothing, and move on, and still tell them your 'offer' still stands. This way, you can move on and they know that if they can't sell, they can call you. If they end up calling you later on and you're STILL interested, ask for more $$$ off! (Just tell them that's their loss. And still, you can always go w/your original agreement. Just don't sign ANYTHING until you verbally agree to terms.)
Now, IMO, NEVER take an Inspector that's been recommended by the Seller. That Inspector is operating in the Seller's interest! It's one thing if the Inspector knows the Sellers (who I'm including the Realtor too) but he wasn't referred to you by them. Sometimes, finding an Inspector just outside the local selling area will get you a more unbiased Inspector. You can always 'protect' you and him by not showing 'them' the Inspector's report. You can be very general in your reasoning as to why you don't want the home.
Now, one more thing. Do you have a Real Estate Attorney??? If not, get one, especially where there are matters such as Riders and Clauses. Unless you're an Attorney, you won't know how to deal w/them. A professional in that line of work is BEST to write them and advise you on them. Ask your friends for references. It's best to find a Real Estate Attorney vs. one that's, ... say, a divorce lawyer and does some real estate 'work'. This way, you've already retained an attorney and you won't have that fee should you need him/her later on. Also, this provides someone working in YOUR interest.
I could go on and on about Inspectors and Attorneys. Bottom line - You need people that are working for YOUR interests; not that of someone else.