I don't know of any algaecides that are so bad you need to drain your pool. If you'll give me the exact algaecide name, I can verify that.
Do NOT drain your pool if it is vinyl or fiberglass; do NOT drain it if its concrete if the ground in your area is saturated with water from heavy rains.
DO consider checking with another pool service -- I can't be sure, till you give me the exact algaecide name, but this sounds like a scam to run up the bill on a new customer, before dumping them. HTH currently makes products that I'd never recommend . . . but I'm pretty doubtful that you need to drain the pool.
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