We often times have to hire/pay a locksmith since the sellers don't even have keys to these houses, etc. 

I would have to get a locksmith to make copies and that is easily $100+.

And keep mentioning how many people are relying on this deadline in conversations - You, your new tenant, their current landlord, the new tenant there, your handyman, locksmith, cleaning crew, etc.

I actually wondered myself if this was the same guy mentioned by @Robert Abasolo in one his videos and in [ Tiny Houses, Interior Design, & Drunk Locksmiths w/ Robuilt - YouTube ] because he acted the same exact way, and used the same expression ("I got you . . .

Can I have a locksmith drill out the locks and enter the property like the bank does in order to winterize and protect the property? 

Locksmiths, 4. 

Landlordlocks uses the same cores/techniques that locksmiths and large buildings use.

. - and that a locksmith has been engaged to change the locks upon their departure. 

Tenants can pay me $50 and I'll bring them a spare key, or they can hire a locksmith for $65.

If it was me and our state laws if no one answered I would show up with lock smith show them my sherrifs recipet and change the locks and leave my card that usually gets a call back.but in some states you cant do anything until the sherrifs deed is actually recorded which can take 30 to 120 days depending on county.. you really need to bone up on the rules there and what you can and cannot do without getting in trouble..