This is a theory first proposed by James Howard-Johnston in his Witnesses to a World Crisis: Historians and Histories of the Middle East in the Seventh Century (2010), which I gather is also the source used in the podcast. Essentially, Howard-Johnston argued that two late Greek sources of the siege, Theophanes the Confessor and Nikephoros, both from the ninth century, based their accounts on a source that was mistaken (postulated to be Trajan the Patrician from the eighth century). Non-Greek sources however did not preserve the same story. Chroniclers following the eighth-century Theophilus of Edessa for example noted a Roman victory in Anatolia in 674 and the insertion of elite troops into Syria in 678, which would have been implausible if the capital itself was under siege. The Greek source, 'Trajan', was writing just after a genuine siege of Constantinople in 717-8 and he had difficulty collating a coherent history of the earlier source-poor period, which perhaps led to his interpolation of an Arab offensive into an Arab siege. Instead of there being an empire under siege, the 670s can instead be seen as a time of Roman military resurgence after the problems of the 660s.

I however have yet to encounter someone who agrees with this. Instead, I've talked to people more inclined to believe that there was an Arab siege of Constantinople in the seventh century. Indeed, there were probably TWO sieges, the first in 654 and the second in 668. 654 is again a very new development, first originating in the late 1990s when the Armenian History of pseudo-Sebeos (c.650s) became recognised as the premier source of the seventh century given its reliability and scope (it had rarely been used before then). For 654, pseudo-Sebeos narrated an Arab attack on the imperial capital after a decisive Arab naval victory over a Roman fleet. Unfortunately for the Arabs, a miraculous storm appeared and destroyed the Arab fleet blockading the city, which led to the Arab army's withdrawal. Further clues can be detected in other sources and this view is now basically accepted by everyone, such as in the narrative histories of Peter Sarris, Empires of Faith The Fall of Rome to the Rise of Islam, 500-700 (2011), and Jonathan Harries, The Lost World of Byzantium (2015). The latter is a new book aimed at the general public, so I heartily recommend it. For a survey of the evidence for 654, see S. O’Sullivan, ‘Sebeos' Account of an Arab Attack on Constantinople in 654’, Byzantine and Modern Greek Studies, 28.1 (2004) and S. Cosentino, ‘Constans II and the Byzantine Navy’, Byzantinische Zeitschrift, 100.2 (2008). Howard-Johnston is also a noted supporter of the 654 siege.

The 668 siege of Constantinople on the other hand is even newer, as it was only posited by Marek Jankowiak a few years ago, which is now published as 'The first Arab siege of Constantinople’, Travaux et Mémoires, 17 (2013). Like Howard-Johnston, he recognised that the evidence provided by the Greek sources is very dodgy, but he demonstrated through a contemporary source that Constantinople was indeed besieged in 668, one that was later transformed by unreliable Greek authors into an epic siege in the 670s. The relevant passage is this:

In addition, I have also the original sealed synodical letter made by Thomas of holy memory, the late patriarch, to Vitalian of holy memory, the late pope of Rome, which, when he wanted to send it to him, he was unable to do so because, as you know, of the long-lasting incursion by the godless Saracens and their presence throughout the two years when he was a bishop.

This alone is not enough, but Jankowiak did a stunning amount of work (the article was 80-pages long!) in uncovering evidence that this indeed happen. Much like the events of 654, crucial contemporary references were often overshadowed by the 'authoritative' Greek sources from much later, so their full importance was not recognised until very recently. However, because this is a very new argument, there isn't much stuff published debating this issue, though it is notable that I anecdotally know of academics who agree with this, whilst Vivien Prigent and Robert Hoyland, two historians whose writings I enjoy very much, have published work that explicitly accepted Jankowiak's conclusions. My own work for my Master's this year dealt with the two sieges of 654 and 668. In fact, I dare say that this is as cutting-edge as it gets in the study of history in academia!

So to answer your question, no, the siege of 674-8 didn't happen. It instead was a mistaken record of a siege in 668. But even that wasn't the first one, as the same had already occurred in 654. However, I am actually very happy that the podcast brought it up, because it is so rare for podcasters to engage with recent scholarship. Howard-Johnston's book is an excellent piece of scholarship and although I disagree with some of his conclusions, it is fantastic that his analysis now has a bigger audience. Pierson's interest in reading what modern historians think about their respective topics rather than just regurgitating the 'traditional' narrative easily makes his podcast one of my favourites. I hope you are enjoying it as much as I am!