However it took them much longer to become a legitimate team than 4 years
As far as I'm concerned, they proved they were a legit team in '79 when they went to the NFC championship game, and they proved it wasn't a fluke by winning the division again two years later. Yes, they had some terrible years before and after that but I think they had organizational issues for years that affected the type of teams they put on the field. That's not my point.
What is my point is the fact that Tampa could call themselves division champs after 4 years of existence, starting from nothing, in an era where free agency as we know it didn't exist and success largely depended on drafting well and executing good trades.
What is also my point is that:
Cleveland went from zero to the playoffs in 4 years in the era of modern free agency.
Jax went from zero to the AFC championship game in 2 years by drafting well and filling in other key positions with solid FA acquisitions. They (and Carolna) had the advantage of it being an era where a lot of teams didn't yet understand cap constraints and as a result, there were a lot of good FAs available back then. However, Jax used the draft to build moreso than Carolina.
Carolina went from zero to the NFC championship game in 2 years using FA as their primary model. Their success was understandably short lived.
The Texans had a boat load of veteran expansion draft selections and have drafted in the top 10 for the last three seasons. The argument about "how can we compete with a team that has been around for 20 years" is weak. If the other teams I mentioned can do it (go to the playoffs in 4 years), so can we. That's my point.
Now, depending on what method you choose to build your team and whether it can be sustained is an entirely different argument. But, that said, Cleveland used the most conservative model of those I mentioned and was still able to "get there" in 4 years.