What a great question; here are my thoughts. The guiding principle is to determine if price or value is your objective.
When you are looking for price, I want to suggest establishing those dollar thresholds for an RFQ (request for quote) when the risk is low and the potential savings are low. Or you are procuring off-the-shelf type products and services. Those quotes ensure you can get the lowest price. To make this work, your procurement team must have a good handle on the products and services. And you must be able to articulate that information in your policy, stating that purchases in those categories above the set limit require the RFQ process. As to where you set that limit, you will need to consider what resources are available to manage the process and balance against the percentage of savings. If the potential savings are not equal to or above the cost of labor to achieve them, it isn't worth the effort.
If you are looking for value, use an RFP of RFI. This approach is best when you have very complex, high-risk products and services. The potential for savings or to negotiate price is also moderate to high. This approach is best when you seek to deliver the product or service in the best way for your company. The price is one of the components vs. the sole objective. Again you will need to identify which categories require an RFP
Based on my experience, that is what I would recommend. However, I am sure there are many ways to approach the situation. I would love to know what our other members would recommend.