Indeed this paper found association between viral load and symptoms - hospitalized patients are shedding less virus
Some flu vaccines come in the form of a nasal spray, which does induce an IgA response. I don't know why COVID vaccines are administered parenterally instead, I'm sure there are good technical reasons. Given the virus's possible neurological involvement, maybe it's a bad idea to deposit it so close to the olfactory bulb.
0: https://en.wikipedia.org/wiki/Immunoglobulin_A#Secretory_IgA
However, the science is the easy part here. It gets complicated once politics comes into play. Pretty much all scientists I've spoken or listened to say that they assume that the spread will most likely be at least severely limited through the vaccines. That's because they don't have any other explanation than a reduced viral load that could explain how the vaccines actually work in preventing sickness, and a reduced viral load would also mean a reduced spreading capability. However, not knowing something does not mean it can't exist: it is not impossible for there to be some unknown way in which these vaccines may potentially prevent sickness without also limiting viral spread. And scientists usually want to be scientifically correct, so they don't go out and declare something as fact that they just carefully "assume" to be the case.
This "known unknown" of effectiveness in limiting viral spread is now actively being used by politics, especially those people coordinating protective measures in governments, as a convenient escape hatch out of a problematic situation. That situation is: how do you explain to someone who just got vaccinated that he/she should still adhere to all the protective measures like universal mask-wearing and limiting personal contacts? Because even though there are rational reasons for doing so (first, the vaccines need two shots and some weeks of time to actually build up protection, and second, if a significant portion of the populace is exempt from all the protective measures and their burdens, this incentivizes the remaining, non-vaccinated part to also "exempt themselves", because constraints enforced on all citizens are much easier to follow than constraints only enforced on half of all citizens) there will be a strong impetus within each individual to not follow protective measures and rules anymore, once vaccinated and thus "protected" from the virus. And besides that psychological effect there's the overarching problem of the constitutional impossibility of enforcing wide-ranging limitations on constitutional rights onto people that is not justifiable anymore, once an individual in question can provably no longer spread the virus. Because of all of this, politicians (rightfully) fear the situation that we're going to be in in a few months time, when a significant part of the populace, but not enough for herd immunity, is already vaccinated. An assumption of a possibility of transmission even with vaccination is a godsend in that situation.
Additionally, for the immune system to respond, the infection has to happen first -- the virus has to spread through the cells of your body. We already know that the people are indeed infectious before their immune system response makes the symptoms. The delay in response has to exist even among the vaccinated people.
So what is sought after is a proof of sterilizing immunity, and there's no such still. I've read that the UK plans to evaluate the evidence for that in the following months by tracking the people who get the vaccine, which sounds good.
Because of this, for practical purposes, you either have to arbitrarily set a threshold at which someone is considered non-infectious, ignoring that it is not impossible for that person to infect someone. Or you have to stop talking in absolutes entirely and just talk about probabilities.
The same is with the possibility that a vaccinated person infects somebody else. There is some point behind which it could be said that some vaccine has "sterilizing immunity" even if some small level of viruses could be present somewhere. For the current vaccine, the question is if the viral load in some point after the infection is decreased at all, and if it is, how much.
At the moment, however, it's simply not known if, in this case, Pfizer vaccine provides sterilizing level of immunity, if, then when, and in which percentage of the vaccinated. At the moment more or less we just know that the vaccinated are less probable to develop symptoms. Efficacy of 95% here means only one of 20 vaccinated develops symptoms when exposed to the virus, so we know that it's also probable that at least 1 in 20, even after being vaccinated, could be able to infect somebody else while being in the "pre-symptomatic" phase (as it is believed by the researchers that one transmits the virus before one's symptoms starts). We also believe that asymptomatic are also able to transmit. We don't know how much the vaccine affects the transmission that could occur when a vaccinated person is exposed to an infected one, and then later comes in close contact with other unvaccinated persons.