It likely would have been, but in the present tense, it would not.
First, a couple of historical notes, Taiwan is considered part of 'Historical China' only under a very recent view of Chinese history. Until very near the end of the Qing, when the island was initially handed over to Japan, it was considered a backwater, and largely left to govern itself. This is one of the reasons Japan's takeover of the island in 1895 enjoyed significant international support at the time--pirates based in Taiwan and the Pescadores (modern day Penghu) were a constant threat in East Asian trade.
Additionally, while the significant Chinese population was largely happy to have a Chinese government back in power after the end of WW2, there was at best a shaky legal basis for the ROC to claim the island (one would need at minimum to consider its initial handover to Japan to be unlawful, to accept the ROC as a successor state, and accept Japan's territory of the entire island as being equivalent to the contained colonies of the Qing) and rule it, and local favor quickly turned against the incoming Nationalists, especially around events like 228.
As the ROC on Taiwan spent its first few decades as a dictatorship under Chiang Kai-shek and the KMT, it was devoted to reclaiming the mainland. Even as this became a pipe dream and western nations began to accept the PRC as the legal representative of China, CKS and the KMT were so strongly opposed that they held on to their claim even as the PRC replaced them in the UN. Had they accepted pleas at this time to declare Taiwan an independent nation, it's quite plausible that the CCP would have accepted it as part of the rise of the People's Republic. CKS would not accept this.
As Taiwan democratized, pro-Chinese sentiment began to fade, and support of Taiwanese sovereignty rose, China's power likewise grew, and it has increasingly used a series of carrot-and-stick tactics to try to keep the possibility of reunification open, and prevent pro-independence forces from becoming too popular in Taiwan. As already mentioned by others, China has threatened (legally and politically) to attack if Taiwan changes its constitution--this has led to concerns not just of changing the name, but even smaller (and more apparently 'obvious') changes such as "acknowledging Mongolia as an independent country" or "formally abolishing Sun Yat-sen's five branch constitutional government".
It's also worth noting that even some Taiwanese Nationalists find the Republic of China name if not desirable, certainly convenient. The Republic of China, like the People's republic, claims significant parts of the South China Sea, as well as other territories that might not fit within a "Republic of Taiwan" such as Kinmen and Matsu, and there is support for this among those who want to see their nation as a strong sovereign, even if it otherwise seems hypocritical.
What the professor is referring to is a bit more nuanced than the idea it sparks: He believes that a complete lack of diplomatic support for the Republic of China while countries continue to acknowledge Taiwan separately from the PRC would strongly encourage the view of Taiwan as an already independent, sovereign nation. What the PRC wants instead is to be seen as the 'rightful China' with the superficial image of Taiwan as a false Chinese Government ruling a very much really-Chinese territory. I've previously heard reports that China has discouraged/declined certain Latin American countries changing their diplomatic recognition in the interest of preserving the status quo (and not hurting relations with a then-friendly Taiwanese government); I do not know the veracity of that, but it would not be unreasonable for them to do so.
In essence, his remarks could be rephrased as "Scant diplomatic relations for the Republic of China may increase, rather than decrease, the People's Republics claims of legitimacy. A Republic of China without any diplomatic acknowledgement may do more to undermine the One China position as a whole than it does to strengthen the PRC's claim to Taiwan".
TLDR: There are many people on both sides who do not want the name changed, even if they don't have serious problems with Taiwan's status quo. Changing the name would, under the PRC's laws (and the corner CCP politicians have painted themselves into), be tantamount to a declaration of war, and would be viewed as provocative even by Taiwan's allies. China does not care about the ROC's claims to its territory or name--it wants them, both to encourage the view of Taiwan as part of China and the 'Taiwan Problem' as an internal dispute, and to generally strengthen the perceived validity of Chinese claims to other territories. Likewise, many of those in Taiwan, even those who do not want reunification or a revival of a Republic of China, do not want to give up certain "Chinese" territories, much less to spark conflict with China.
