I very much concur with what you will find in the other answers, but I think there are a couple points worth addressing or emphasizing as well as a need to answer the precise question.
Yes, it is Possible and Normal
To answer your exact question of whether it is a possible thing to do and a normal thing to do, it is simply an objective fact that the world of this hobby is full of many groups integrating new members/characters at some point or other if not frequently. Groups lose someone, never had enough people to begin with, or just have a person they want to play with. To cite personal experience I've had it happen, albeit usually early on, in every group I've had that did not start as a close friend group and lasted more than two sessions. One of mine that is a friend group is planning to incorporate a new member at the next good juncture, in part precisely because they are a housemate of the player whose house we play at, as well as it being a friendly thing to do and something that should help round out the party.
Exactly how common it is in the hobby overall is something nobody could really know for certain, but it is not uncommon by any means.
But Don't Play With People Who Don't Want You
However, I think it is important to emphasize, that a group that does not want you there is not a group you are likely to enjoy playing with. Even if your very best friends, if some of them are dead set against it then human nature would be for at least some of them to notice the problems with your play and playstyle disproportionately to the positive contributions you make. This doesn't necessarily mean it would not work out if they gave you a shot, simply that there is a high likelihood of it not being a good experience.
Never play where you are unwanted, it will rarely be a pleasant experience. The conventional wisdom is that no D&D is better than bad D&D, though that is of course a matter of opinion. In any case unless you are in a very small community, very shy about new people, or simply not interested in a group that is not already your friends, you should have no trouble finding other groups that want a new player and statistically some of them should mesh with your play.
And Don't Hold Turning You Down Against Your Friends
There may be some underlying personal issue with you or how they anticipate you would be as a player. Or there may be some misconception of how incorporating a new player would go that has unreasonably prejudiced them against it. However there are also several fairly sensible reasons that have nothing to do with you why they would not want another person in the group.
The absolute best and most common would be that one more person is simply too many for the group. Every additional player means less time other characters can be taking an active, or at least the most active, role in things, and longer they have to wait for their combat turn. It is also more work for the DM, and still more the more carefully they tailor the adventure to the characters.
The next best would probably be that they simply have a group dynamic they are very happy with and are afraid to mess with success. A really good group dynamic is a precious thing, which some will guard carefully. If you were a stranger they could probably try you out, see if you mesh, and drop you if it doesn't work out without ever needing to worry much about it. When you're a friend who lives with them though it's a very different situation where rejecting you after a session is much more uncomfortable than preemptively excluding you.
They may really not want to deal with showing a new player the ropes, which is not something I personally can remotely identify with, but is a perfectly reasonable perspective.
The stated reason of "not being fair to other players" might not generally hold up if you agree to start at level one on an experience point system, since they will be dealing with experience point numbers that level a low level character up rapidly. That said, most groups share out XP evenly to all players or at least all players at the session, which means that everyone else potentially levels slower by having another person. Still XP is the DM's prerogative in how they structure XP opportunities or just in how much they give out by fiat, so this is mainly a "more work for the DM" problem.
Meanwhile if you are leveling at milestones this would probably require you getting milestones much more often than everyone else for a while, which may be a bummer for other characters. In either system they may find it unfair that you get to level so fast when it was such a slog when they did it. Both these issues veer into the realm of an 8 year-old "he gets a cookie so give me a cookie" conception of fairness. Nevertheless if that's something they need to enjoy this recreational experience which they do solely for fun and have no obligation to participate in, then such is life.
Some groups may have a preference against having varying experience levels in the group, which legitimately may force a DM who calculates encounters very carefully to have to worry about balancing them more. They may worry that between starting low and being a newer player you won't be able to pull your weight. I'd argue that any player contributes something unique to a game, that unless you have a very large party there are always unfilled niches that can be useful to have someone specialize in, and that a new character provides potential to explore new aspects of their own characters and get cool new adventure hooks. Nevertheless this may not be enough to make it worthwhile.
If they are in tier three or even tier four of play (as in those over level ten, to gloss the jargon) it is entirely likely that a very low level character would be downed constantly with the enemies they face. You would be downed almost any time you were hit by many of the enemies for the first couple levels, and then they are burning healing spells and potions constantly to keep you in fights, and potentially having to raise you from the dead. At the very beginning there is a substantial risk of being hit with so much damage that you die outright on a fairly regular basis (a single fireball with a high die roll will outright kill any first level character under normal circumstances).
On a final note you mention that characters die and need to be replaced in their game "nearly every month". This is not at all the case for many groups, my own included, but it would tend to indicate an at least somewhat gritty blood-and-guts campaign style (the norm at the beginning of the hobby, less common in later editions). Under such circumstances a low level character is less likely to be viable to play with high level characters and where a newer player's mistakes are more likely to have dire consequences for themselves and others.
At the end of the day, while they may be entirely offbase (and their stated reason is one of the weaker one) they are in a better position to accurately judge these things than you or any of us because they are the ones who actually have experience with their table.
But, if they ever do oneshots, where the fairness thing is irrelevant and which often happen when groups are short a player or two anyway, they may feel very different about you joining. And should that go awesome enough their stance may well change.