What are the pros and cons of having a business model - snatching other businesses existing customers, especially a month or two before they renew with them? Is there any business term for what I have just described above?
It depends on the sector as to if you really want to do this.
In the British fuel and heating market (Gas and Electric) this is the way the entire market runs. The same can be said for car insurance and home contents insurance. It's the norm.
This would be unlikely to work with Gym membership and is unlikely to work where the person paid some upfront fee in addition to their subscription or whatever they are paying.
Generally snatching customers is a pricing game. Being a little bit cheaper is often the biggest (or only) selling point as they have to risk changing from what they know to what they do not (you). Seth Godin (check him out) calls this the race to the bottom.
You are going to need a well motivated cold calling sales team and in all likelihood you are going to be shedding some profit in commission to these guys and gals.
Be prepared for your competitors to fight back and try to do the same to you. If you are all SMEs then you might find the biggest cost is social in that the other owners are not going to be keen to buy you a drink...
If this is going to be your approach do it hard fast and at full steam and don't be afraid of putting smaller competitors out of business or simply buying them up to get their customers.
Also look to the cost to acquisition and profit ratio. It is very easy to spend a few pennies more per customer than you ever earn from them. The bit in the middle between cost of getting them and cost of serving them is your profit. Keep it big.
I would say this is a tactic but not a business model. You can use many tactics on many aspects to support a model, but a model can't be based on tactics.
Besides I don't believe there is a legit method to know which customers have expiring plan on competitors. Maybe you know, but a little caution is still needed for it may be grey.deleted because OP mentioned his legit method in comment. My two cents about the tactic:
Pros: Effective. Time and cost saving.
Cons: Labour intensive. Limited audience