Regarding the small Veblen ordinal, Rathjen and Weiermann gave an analysis of theories in that range of strength in Proof-theoretic investigations of Kruskal's theorem. Working over a reasonable base theory ($ACA_0$, the second order version of Peano arithmetic), both the theory $ACA_0+$Kruskal's theorem and a somewhat technical theory ($ACA_0$ plus $\Pi^1_1$ reflection for $\Pi^1_2$-BI; $\Pi^1_2$-BI is induction along internally well-ordered sets for $\Pi^1_2$ formulas, and $\Pi^1_1$ reflection means that the statement itself doesn't hold, but all its $\Pi^1_1$ consequences do). (I'm not sure if any of their theories are exactly the large Veblen ordinal.)

The more general question seems too vague to answer. As the Rathjen and Weiermann article shows, there are an awful lot of reasonable-ish theories out there.

Regarding the small Veblen ordinal, Rathjen and Weiermann gave an analysis of theories in that range of strength in Proof-theoretic investigations of Kruskal's theorem.¹ Working over a reasonable base theory ($ACA_0$, the second order version of Peano arithmetic), both the theory $ACA_0+$Kruskal's theorem and a somewhat technical theory ($ACA_0$ plus $\Pi^1_1$ reflection for $\Pi^1_2$-BI; $\Pi^1_2$-BI is induction along internally well-ordered sets for $\Pi^1_2$ formulas, and $\Pi^1_1$ reflection means that the statement itself doesn't hold, but all its $\Pi^1_1$ consequences do). (I'm not sure if any of their theories are exactly the large Veblen ordinal.)

The more general question seems too vague to answer. As the Rathjen and Weiermann article shows, there are an awful lot of reasonable-ish theories out there.

¹Rathjen, Michael; Weiermann, Andreas, Proof-theoretic investigations on Kruskal’s theorem, Ann. Pure Appl. Logic 60, No. 1, 49-88 (1993). ZBL0786.03042.

Regarding the small Veblen ordinal, Rathjen and Weiermann gave an analysis of theories in that range of strength in Proof-theoretic investigations of Kruskal's theorem. Working over a reasonable base theory ($ACA_0$, the second order version of Peano arithmetic), both the theory $ACA_0+$Kruskal's theorem and a somewhat technical theory ($ACA_0$ plus $\Pi^1_1$ reflection for $\Pi^1_2$-BI; $\Pi^1_2$-BI is induction along internally well-ordered sets for $\Pi^1_2$ formulas, and $\Pi^1_1$ reflection means that the statement itself doesn't hold, but all its $\Pi^1_1$ consequences do). (I'm not sure if any of their theories are exactly the large Veblen ordinal.)

The more general question seems too vague to answer. As the Rathjen and Weiermann article shows, there are an awful lot of reasonable-ish theories out there.

Regarding the small Veblen ordinal, Rathjen and Weiermann gave an analysis of theories in that range of strength in Proof-theoretic investigations of Kruskal's theorem. Working over a reasonable base theory ($ACA_0$, the second order version of Peano arithmetic), both the theory $ACA_0+$Kruskal's theorem and a somewhat technical theory ($ACA_0$ plus $\Pi^1_1$ reflection for $\Pi^1_2$-BI; $\Pi^1_2$-BI is induction along internally well-ordered sets for $\Pi^1_2$ formulas, and $\Pi^1_1$ reflection means that the statement itself doesn't hold, but all its $\Pi^1_1$ consequences do). (I'm not sure if any of their theories are exactly the large Veblen ordinal.)

The more general question seems too vague to answer. As the Rathjen and Weiermann article shows, there are an awful lot of reasonable-ish theories out there.