What is the status of the Raymo & Ruddiman idea that Tibet cooled the Earth?
The Tibetan Plateau uplift is still generally considered as playing a role in the Neogene cooling through the process you explained in your question (see for instance the seminal Zachos et al. 2001 and more recently Garzione 2008).
It is probable that other phenomena played a role as well such as the diversification of diatoms (which are, today, the most efficient biological carbon sink, e. g. Goldman 1993; Ragueneau et al 2000) around 15 Ma (Lazarus et al 2014 [Disclaimer: I am a co-author of the latter paper] ) or changes in the ocean circulation (specifically an increase in the proportion of bottom water originating from the Southern Ocean; Flower &amp; Kennett 1994).
That being said, the cooling in question here would be the one starting at ca. 15 Ma. However the Cenozoic cooling trend started way earlier, in the Early Eocene (after the Early Eocene Climatic Optimum, aka EECO), with ice on the eastern Antarctic
continent as early as the Late Eocene / Early Oligocene (see again Zachos et al 2001).


Sources:  

Flower, B. P. &amp; Kennett, J. P., 1994. The middle Miocene climatic transition: East Antarctic ice sheet development, deep ocean circulation and global carbon cycling. Palaeogeography, Palaeoclimatology, Palaeoecology, 108: 537-555.  

Garzione, C. N., 2008. Surface uplift of Tibet and Cenozoic global cooling. Geology, 36: 1003-1004.

Goldman, J. C., 1993. Potential role of large oceanic diatoms in new primary production. Deep-Sea Research I, 40(1): 159-168.

Lazarus, D., Barron, J., Renaudie, J., Diver, P., &amp; Türke, A., 2014. Cenozoic Planktonic Marine Diatom Diversity and Correlation to Climate Change. Plos One, 9(1): e84857.


Ragueneau, O., Tréguer, P., Leynaert, A., Anderson, R. F., Brzezinski, M. A., DeMaster, D. J., Dugdale, R. C., Dymond, J., Fischer, G., François, R., Heinze, C., Maier-Reimer, E., Martin-Jézéquel, E., Nelson, D. M., Quéguiner, B., 2000. A review of Si cycle in the modern ocean: recent progress and missing gaps in the application of biogenic opal as a paleoproductivity proxy. Global and Planetary Change, 26: 317-365.  

Zachos, J., Pagani, M., Sloan, L., Thomas, E., Billups, K., 2001. Trends, rhythms, and aberrations in global climate 65 Ma to Present. Science, 292:686-693.