Can you ritual cast the Phantom Steed spell multiple times and in practice just re-summon them while you are traveling? So for example, could a Wizard ritual summon a horse, three times in a row. And then just focus on re-summoning them while he's riding one of them. The spell lasts 1 hour, ritual casting it 3 times would take 33 minutes. So he should have more than enough time to summon new horses for himself and his party before any of the individual horses disappear. Is this correct?
There is no rule stopping you from doing this
The relevant aspects of phantom steed are a casting time of 1 minute, the ritual tag, the duration of 1 hour and absence of a concentration requirement.
Ritual casting a spell means the casting time increases by 10 minutes, to 11, with no other requirements added.
Certain spells have a special tag: ritual. Such a spell can be cast following the normal rules for spellcasting, or the spell can be cast as a ritual. The ritual version of a spell takes 10 minutes longer to cast than normal. It also doesn't expend a spell slot, which means the ritual version of a spell can't be cast at a higher level.
— Player's Handbook, p. 201
Note that casting a spell with a casting time of more than 1 action requires your concentration; fortunately we don't need our concentration to maintain phantom steed and so can use it on the new instance.
There is also no rule saying you can't have the same spell cast multiple times, or said another way the rule on combining magical effects does not apply as it reads:
The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect—such as the highest bonus—from those castings applies while their durations overlap.
— Player's Handbook, p. 205
However we are not trying to combine the effects, but rather have them benefit separate creatures, or the same creature at different points in time. (If you are trying to ride two horses simultaneously, or heavens forbid stack them, you might get a problem with this rule)
There are also no rules in the spellcasting (PHB 201), ritual casting (PHB 201), or Mounted Combat (PHB 198) sections of the PHB which prevent ritual spellcasting while mounted. The only exception would be if your DM decided that riding on a mount qualified as a sufficient environmental phenomena to threaten your concentration (PHB 204).
So, with multiple castings you can have up to 5 steeds active at the same time, giving yourself a 5 minute break every hour to relax and of course switch to your new phantom steed.