So we’ve got the basics down. We all carry mutations in genes that are harmless by themselves, but if you get two broken copies of the same gene bad things happen. The answer to your question has to do with probabilities.

Let’s say you have two populations. In the first there are 100 genes with recessive lethal mutations, each at 1% frequency in the gene pool. This means that on average there are 2 recessive mutations per individual (because each individual has 2 copies of 100 genes with a 1% rate of being mutant). The probability of an individual receiving 2 copies of a single recessive mutation assuming completely random shuffling is 100 * 0.01^2 (one hundred chances of getting two copies), or 1%.

Now let’s look at a population with only 10 recessive lethal mutations, but that are now at a frequency of 10% in the population. On average there are still only 2 recessive mutations per individual, but now if we calculate the chances of a person being born with two of the same mutation we get 10 * 0.1^2 which is 10%.

The two populations have the same total number of recessive lethal mutations, but problems arise more frequently in the second population. So while a populations bottleneck will remove the vast majority of genetic issues, the ones that remain will run rampant in the remaining population.

To further illustrate how this represents a population bottleneck, imagine if you take 5 individuals from population 1 and use them to found a new population. Each of these individuals brought 2 different mutations from the original pool for a total of 10, and in the starting population there is 1 mutant copy out of 10 total (5 people, each have 2) so each mutation is at a 10% frequency. Now we have population 2.

To move away from the hypothetical and address your situation directly. It is estimated that the average person carries ~5 recessive lethal mutations. With a founder population of 6, that means you can expect ~30 total recessive lethal mutations. Each of these would exist in your population at a frequency of 1/12 or 8.3%. The first few generations would not show any issues as you are still breeding completely unrelated individuals, but once the bloodlines are sufficiently mixed each child will have a 30 * (1/12)^2 = 20.83% chance of inheriting two recessive alleles and dying from a genetic defect. This chance will gradually decline over time as the children that die won't pass on those mutant alleles so the bad genes will slowly be purged from the population. Essentially for many generations the birth date will be fairly low, but will slowly recover.

It's also important to keep in mind that the above only covers recessive lethal mutations. Plenty of recessive mutations are non-lethal and would also be frequently seen in the offspring of this population. Expect a lot of deformities and other weird stuff.