Hmm. You have been totally honest but this is still somewhat misleading.
You’ve buried the lede, which is that when you say
you are referring to the average time it takes a free player to get an HOTM, not the time it takes an average free player to get an HOTM. There is a big difference. I think you probably ought to be upfront about this.
You have simply calculated 1/p (which is 1/0.13, which is about 77), and combined this with your gem costs (which are correct) and your assumed rate of gem accumulation (which is a bit low, I think) to come to your two year “average”. Again, I think it is perhaps a little misleading not to be plain about this.
So your two year “average” represents the expected time to summon an HOTM. It’s the mean time across all players, but of course the distribution is not symmetrical - it is heavily right skewed by those players who are very unlucky.
So we can also say, using your own assumptions, that we would expect considerably less than half all free players to take two years or more to summon an HOTM (37% if we’re being precise), while we would expect that most free players (63%) would take less than two years, and fully a quarter of free players (26%) would summon multiple HOTM within 2 years.
As I noted above, it will take only 53 summons for an F2P player to be as likely to have received one or more HOTM as none. (To be clear, this is based on a simple binomial distribution.) Based on your (conservative) gem accumulation assumption of 850 per month, this gives a median time to summon at least one HOTM of about 16 months. I think this is a considerably more appropriate way to calculate a typical player experience. (Although we would expect 15% of F2P players to summon multiple HOTM in that time.)
And of course, if we increase your expected gem accumulation rate from your pretty conservative 800-900 per month then our time estimates will fall even further. But let’s leave this part alone: perhaps it’s healthier to be a bit relaxed about the game, take things a little easy, and not feel compelled to grasp every gem that is available.