Why are most of deleterious mutations recessive in nature? I understood that if it's recessive then one reason may be that the mutant gene doesn't code for a functional protein and so there is no phenotype to express. But why is it recessive in the first place?
It is a bit ironic that you phrase your question with 'in the first place'. There is a parsimonious explanation for the effect: Dominant deleterious variants are quickly removed by purifying selection whereas recessive deleterious variants can "hide" in individuals that carry a dominant neutral/advantageous variant.
You can think about it this way: In heterozygous carriers (in which the other variant has a neutral or positive fitness effect), a recessive deleterious variant has no (strong, depending on the dominance coefficient) negative effect on fitness and therefore, in those heterozygotes, purifying selection will not remove the variant (efficiently, again depending on the degree of dominance) - it can only act on homozygous carriers of the recessive variant. A consequence of that is that the recessive deleterious allele usually segregates at low frequency with most carriers being heterozygous.
In short: This probably has little to do with there being more recessive deleterious variants per se (even though that might also be the case due to the mechanisms you described in your question). In the snapshot we are looking at when we investigate genetic data, the likelihood of finding dominant deleterious alleles is way lower than finding recessive deleterious alleles as the latter usually just persist for a way longer period of time.