In reference to the question of whether or not Jesus actually used the word synagogue rather than church, it is highly unlikely in my humble opinion for two reasons. First of all, my copy of Lamsa’s translation from the Aramaic of the Peshitta says "church." Secondly, the word "ekklesia" is a combination of 2 words: meaning (literally) out of from inside of and called. Thus, if is a reference to the body of Christ which is made up of people who are called out of this world into Him. Thus, "church" is a reference to PEOPLE rather than a PLACE or a building, as is the case with "synagogue." Consequently,the Peshitta and the Greek N.T. are in agreement both verbally and conceptually.
One thing I can say for sure is that Jesus neither said "church" nor "ekklesia," as church is an English word and ekklesia is a Greek word and Jesus was speaking Aramaic. What you find in an English translation of the Peshitta also does not tell you what Jesus said. It simply tells you how a particular translator happened to translate the Aramaic word used in the Peshitta translation. Besides, we know that the Peshitta is a translation from the Greek. Therefore, the only thing we have to go on is what the original in Matthew has for Matthew 16:18. My Greek interlinear has Jesus saying on this rock I will build my ekklesia. What does this tell us? It tells us that when Matthew made his own personal translation of what Jesus said in Aramaic, he felt that the best translation of what Jesus said in Aramaic was ekklesia. It is interesting but true that even though Jesus spoke in Aramaic, the Aramaic versions of the New Testament are less reliable at telling us what Jesus said than the Greek, because the Peshitta is an Aramaic translation of a Greek translation from the Aramaic!!! Does that make sense to you? When you read Lamsa, you are reading an English translation of an Aramaic translation of a Greek translation of an Aramaic word. It is a bit of a stretch to make a strong argument from this. So, what did Jesus actually say in Aramaic? The answer is that we simply do not know and we have no way of knowing, and the Peshitta is no better at helping us that the Greek New Testament.
Secondly, you are obviously aware that the word church (ekklesia) in its original meaning did not refer to a building. That is true. That alone does not mean that Jesus did not say the Aramaic equivalent of synagogue, as the word synagogue does not necessarily refer to a building, any more than does the word ekklesia. In fact, in Revelation 2:9 and 3:9 we read about a Synagogue of Satan, which is definitely not a reference to a building or a place. The word synagogue in modern English may well mean a building of Jewish worship, as does the English word church. However, the Aramaic word for synagogue meant the people, not the building, as with the Greek word ekklesia. By the way, in order to support my contention, the word synago in Greek means assembly. This supports the idea that Jesus may very well have said the Aramaic word for synagogue when he spoke as recorded by Matthew.
Anyway, believe it or not, I do not disagree with you!!! It is just that I do not think you can say authoritatively that Jesus did not use the word synagogue in the discourse in Matthew 16:18. We should just say what we think is likely and leave it at that. I am curious. There must be something behind your adamant opinion. Is there some sort of point you are trying to make? I am not sure that there is any important doctrinal point to be made on whether Jesus said "I will build my church" or "I will build my synagogue," given that the Hebrew idea of synagogue is quite similar to the Greek idea of ekklesia. Both mean, in the broad sense, a group of people called out by God.
By the way, I have the Aramaic of Matthew 16:18 from the Peshitta below:
18 ܐܦ ܐܢܐ ܐܡܪ ܐܢܐ ܠܟ ܕܐܢܬ ܗܘ ܟܐܦܐ ܘܥܠ ܗܕܐ ܟܐܦܐ ܐܒܢܝܗ ܠܥܕܬܝ ܘܬܪܥܐ ܕܫܝܘܠ ܠܐ ܢܚܤܢܘܢܗ
Another FYI. The Hebrew word for synagogue is hakneset meaning congregation, while the Aramaic word is knista. Today, the assembly/Parlaiment in Israel is known as the Knesset. So synagogue means congregation or assembly, which certainly fits the context of what Jesus told Peter in Matthew 16:18. In fact, if you think that Jesus said "church" in Matthew 16:18, what Aramaic word would you propose that he used? Again, I am no expert, but I do not know of an Aramaic word closer to the Greek ekklesia than the word knista. There is another Aramic word atra which is translated as place or (religious) site. This word is sometimes translated as synagogue, but it is less likely Jesus used this word for the reason you mention above, which is that it refers to a place rather than a people, whereas knista (synagogue) means congregation or assembly.