A generalization of strong primes

Regarding any $\hat\theta$ for which the prime numbers sequence has finitely/infinitely many terms, consider one which has only finitely many terms. There would then exist a prime index $m$ for which all $n \gt m$ gives

$$p_n \le \hat\theta\, p_{n-1} + (1 - \hat\theta)p_{n+1} \tag{1}$$

Using the standard definition of prime gaps of

$$g_n = p_{n+1} - p_{n} \tag{2}\label{eq2A}$$

we then have

$$\begin{equation}\begin{aligned} p_n & \le \hat\theta(p_{n} - g_{n-1}) + (1 - \hat\theta)(p_{n} + g_{n}) \\ p_n & \le \hat\theta p_{n} - \hat\theta g_{n-1} + (1 - \hat\theta)p_{n} + (1 - \hat\theta)g_{n} \\ \hat\theta g_{n-1} & \le (1 - \hat\theta)g_{n} \\ g_n & \ge \left(\frac{\hat\theta}{1 - \hat\theta}\right)g_{n-1} \end{aligned}\end{equation}\tag{3}\label{eq3A}$$

If $\hat\theta \gt 0.5$, then $\frac{\hat\theta}{1 - \hat\theta} \gt 1$, so \eqref{eq3A} shows the prime gaps are strictly increasing for all $n \gt m$. However, this contradicts that there are infinitely many prime gaps of at most $246$ (e.g., see Bounded gaps between primes). If $\hat\theta = 0.5$ instead, then $\frac{\hat\theta}{1 - \hat\theta} = 1$, so the prime gaps are non-decreasing, but this is also not possible since prime gaps can become arbitrarily large and, thus, must decrease later to become the smaller prime gaps, e.g., those at most $246$.

This means the original assumption of there being only a finite number of primes in the sequence must be incorrect, i.e., there are infinitely many $\theta$-strong primes for any $\hat\theta \ge 0.5$.

Update: The PDF version of the arXiv article On the ratio of consecutive gaps between primes, near the bottom of page $8$, states about Erdős

He mentioned 60 years ago [Erd5]: "One would of course conjecture that

$$\underset{n \to \infty}{\lim \inf}\frac{d_{n+1}}{d_n} = 0 \; \text{ and } \; \underset{n \to \infty}{\lim \sup}\frac{d_{n+1}}{d_n} = \infty \tag{2.2}$$

but these conjectures seem very difficult to prove." Based on a generalization of the method of Zhang [Zha] the author proved (2.2) in [Pin2]

where [Pin2] is

J. Pintz, Polignac numbers, conjectures of Erdős on gaps between primes and the bounded gap conjecture. arXiv: 1305.6289 [math.NT] 27 May 2013.

i.e., here.

Note their $d_n$ is the same as $g_n$ in \eqref{eq2A}. If the first part of (2.2) is true, then no matter how close $\frac{\hat\theta}{1 - \hat\theta}$ is to $0$, \eqref{eq3A} cannot always being true for all $n \gt m$ for any integer $m$. This means there are infinitely many $\theta$-strong primes for all $0 \lt \hat\theta \lt 1$.

I haven't read the article or references to try to verify the validity of the claim. However, note the article seems to be basically very similar to at least part of the Springer Link book On the Ratio of Consecutive Gaps Between Primes, but I haven't paid to get a chapter or the entire e-book, or a physical copy of the book, to check on this.

As requested I put my comment as a (partial) answer:

Since we know there are infinitely many consecutive primes with finite gap $C$ (smallest proved $C$ as of now seems to be $246$), any $\theta$ for which $2\theta >(1-\theta)C$ works for the $p_n$'s precisely the primes for which $p_{n+1}-p_n \le C, n \ge 2$ since then $p_n-p_{n-1} \ge 2$ so:

$\theta(p_n-p_{n-1}) \ge 2\theta > C(1-\theta) \ge (1-\theta)(p_{n+1}-p_n)$

For $C=246$ we get that $123/124 < \theta <1$ satisfies the requirement of the OP with infinitely many $p_n$'s