I have been making offers for clients and getting rejected left and right with 60-100k over asking.

If you don’t  know their new address their old address (your rental) is where you send the demand letter and have them served.

I had a similar idea by attaching myself to an real estate agent so that they can refer me to clients buying properties to rent through Airbnb, but I like your idea even better.

This isn't "first come first serve" it's all about placing the best tenant for the property into the property.

Second, nothing will decide my success as well as my quality of tenants, and while it's one thing to let someone fix a toilet, it's another to let someone decide who your clients are going to be. 

I recently made a connection with a client who is a newer real estate investor.

At least two of them have done kitchen/bath remodels for me personally in addition to serving dozens of our clients over the years.

My clients currently are either cash buyers are working with Hard money.

Looking to work part time and I already have a client who’s ready to buy ASAP and I don’t want to lose him.

No one has the ability to tell you what will happen tomorrow or next month in real estate, but depending on how you answer the questions above may affect the advice I would give a client.