FWIW I've had experience with 3750's (3750G, and then later 3750E/3560E) at scale in a TOR setup; initially with L2 port-channels/GLBP (variants of 2x1G and 2x2G and the rare 2x4G for db racks) and then with L3 to the TOR (went with this for 3750E/3560E and 10G to the core). I'm talking thousands of them. We only saw issues with buffers for the most bandwidth intensive services, and at that point we were looking at 10G to the host anyway (and dense pizza boxes with 24-48 SFP+'s).
Whether or not you're going to be able to prove anything to management is really going to depend on the application and you doing your homework on what the requirements of your design and the application are, and knowing exactly what the specifications are of the application, as well as the expected growth velocity of it. Set up a design review with your management chain as well as the primary owners/customers of the network.
Management wants to see data, and if you don't have the resources to fully test the box (come up with a test plan, connect it up to some traffic generation hardware, fully scope it out and stress test it to the design spec, etc) this is going to be hard to do. They're not going to be impressed with anecdotal evidence, and finding this kind of hard data may prove difficult, since I'm sure that folks publishing this kind of thing would violate all kinds of NDA's.
Everyone else that's posted an answer to this has outlined the 3750 platform's "problem areas" pretty nicely: stacking and weird failure modes inherent to it, buffer sizes, etc. There's also this question that outlines the issues with gathering SNMP statistics on output queue drops - the buffers are shared across the ASICs, so any stats you get with this via SNMP are going to be the same for specific port ranges (this could be one sticking point you could bring up with your management chain).
To summarize, I'd say that the 3750/3560 would be "fine" for most deployments, even at somewhat large scales. Avoid stacking them if you can, but I'd say that it's not too horrible to do this in very small and manageable quantities.