Karger's min cut and tips on bounding nonlinear recurrences
First of all, let me explain why your estimate is lossy: when $P_i$ becomes small, $P_{i-1} - P_{i-1}^2/4$ is very close to $P_{i-1}$, and in particular, it doesn't decrease exponentially.
Since the function $x - x^2/4$ is monotone in the range $[0,1]$, if you solve the corresponding recurrence $Q_i = Q_{i-1} - Q_{i-1}^2/4$ (with base case $Q_0 = 1$), then $P_i \geq Q_i$.
Let us write the recurrence relation as $Q_i - Q_{i-1} = -Q_{i-1}^2/4$. This is a difference equation which can be approximated by a differential equation $q' = -q^2/4$, with $q(0) = 1$. Using standard techniques, you can determine the solution, which is $q(i) = 4/(4+i)$. This suggests that $Q_i \approx 4/(4+i)$, and so hints that $4/Q_i$ is a useful quantity to look at.
Here is another way to estimate the recurrence. Write the recurrence as $$ Q_i = (1-Q_{i-1}/4) Q_{i-1}. $$
In a range of $i$ where $Q_i \approx \theta$, it takes $\Theta(1/\theta)$ steps to reduce $Q_i$ by a factor of $1/2$. In particular, after
$$
\Theta(1+2+\cdots+2^{\ell-1}) = \Theta(2^\ell)
$$
steps, we reach $Q_i \approx 2^{-\ell}$. This suggests that $Q_i$ scales like $\Theta(1/i)$. With a little bit of work, this argument can be made completely formal, though it is hard to get a hold on the exact constant factor in this way.