During my modern Chinese history course, my professor mentioned a major reason that the British and French forces won the Second Opium War was because the Qing government was tackling other rebellions simultaneously and could not bring their full numbers to bear on the Western forces.
In essence, I am curious if this is true and if such numbers would have been enough to overcome Western technology at the time, as said prof also stated the British Army was often well trained and equipped, but poorly led?
Your teacher is right in part: the Qing simply were not that concerned with the British and French, who did not represent an existential threat to the empire, compared to the Taiping, who did. For just one illustrative example, over 100,000 Qing troops were killed, wounded or captured during the Taiping campaign to relieve the siege of Nanjing in early 1860, whereas 50,000 were deployed against the British and French during their march to Beijing in August. And that army around Nanjing (whose total numbers were several tens of thousands higher) was only one of many Qing forces fighting the Taiping at the same time. Now, to be sure, those 50,000 troops were largely Mongol and Banner troops, the core standing elements of the Qing army, and outnumbered the British and French some 5-to-1 during the final phase of the Arrow War. But even so, it is fair to say that Qing energies were very much focussed elsewhere.
At the same time, the experience of the First Opium War shows that the Qing would have had a hard time either way. The European armies were not the biggest problem, but rather their navies, which allowed land forces to reposition rapidly, attacking vulnerable positions before they could be reinforced, avoiding difficult engagements, and sailing away where Qing vessels, being slower and suited only to coastal waters, could not pursue. Of course, this meant that these armies could not advance far inland, but they didn't have to. The British and the French were after trade concessions and some very limited territorial changes, not the destruction of the Qing state. All they had to do was make continued resistance less attractive than an immediate peace. The ongoing Taiping War and the immense expenditures required to suppress it would have made such a proposition much more realistic.
This past answer focusses on the 1839-42 Opium War and covers the Arrow War largely as an afterthought, but the basic dynamics were similar enough. But if you have follow-ups or requests for further detail, I'm happy to provide.