This is simply not true, the Early effect is present in the common-base configuration as well.
A good example is the cascode configuration, where the cascode transistor is in a CB configuration. Rout depends on the early effect, however it is reduced because of negative feedback. Additional current through a cascode stage decreases Vbe of the cascode transistor, which reduces the impact of the Early effect.
Just redrawing your schematic slightly and adding some labels:
simulate this circuit – Schematic created using CircuitLab
Just use KVL to start, following around through the base:
$$\begin{align*}
V - I_{Rc}\cdot R_c - I_{Rb}\cdot R_b - V_{BE} - I_E\cdot R_e &= 0
\end{align*}$$
If the BJT is in its active region where your value of \$\beta=175\$ applies (and you'll know one way or another, soon enough), then it also follows that:
$$\begin{align*}
I_{Rc} &= I_C+I_B=I_E \\
I_{Rb} &= I_B \\
I_E&=\left(\beta+1\right)\cdot I_B
\end{align*}$$
Applying those to the original equation, we get:
$$\begin{align*}
V - I_E\cdot R_c - I_B\cdot R_b - V_{BE} - I_E\cdot R_e &= 0 \\
\\
V - \left(\beta+1\right)\cdot I_B\cdot R_c - I_B\cdot R_b - V_{BE} - \left(\beta+1\right)\cdot I_B\cdot R_e &= 0 \\
\\
V &= V_{BE} + I_B\cdot\left[\left(\beta+1\right)\cdot\left(R_c + R_e\right) + R_b \right] \\
\\
I_B &= \frac{V - V_{BE}}{R_b+\left(\beta+1\right)\cdot\left(R_c + R_e\right)}
\end{align*}$$
And that pretty much cracks the puzzle.
(It ignores little-re, which might have an impact in some cases but probably doesn't, here. It's impact is probably below 1%. But you could work it back into the equation later, if it matters to you.)
At this point, given your values and using \$V_{BE}=700\:\textrm{mV}\$, I get:
$$\begin{align*}
I_B&\approx 10.1\:\mu\textrm{A} \\
I_E&\approx 1.78\:\textrm{mA}
\end{align*}$$
So, I'd estimate:
$$\begin{align*}
V_E&=I_e\cdot R_e\approx 1.78\:\textrm{V} \\
V_C&=10\:\textrm{V}-I_E\cdot R_C\approx 6.44\:\textrm{V} \\
V_B&=V_E+V_{BE}=V_C-I_B\cdot R_B\approx 2.48\:\textrm{V}
\end{align*}$$
Since \$V_{CE} \gt 1\:\textrm{V}\$, the BJT is in its active region and the value of \$\beta=175\$ can be considered to have applied, now that we can check it out. So it is fine to stop at this point and consider the question answered well enough.
Little re is because the base-emitter junction's thermal voltage can be treated as a tiny "battery" at the tip of the BJT emitter. It's always got the thermal voltage just sitting there, which at room temperature will be around \$26\:\textrm{mV}\$. Given a current through it (\$I_E\$), you can turn that into an equivalent resistance. This is called a lot of things, but in just talking I say "little re." In this case, \$re\approx 15\:\Omega\$.
This value adds to \$R_E\$ in the above calculations. Working it out, I find that it impacts the estimated current values by about 0.3%.
Just a note.
Best Answer
Why does adding Rb2 increase stability with respect to variations in Beta
That's not difficult to see. Adding Rb2 would "steal" some current from the base of the NPN, so to prevent that we decrease the value of Rb1 such that it provides extra current.
Now if the base current is 1 uA and we make 100 uA flow through Rb1 that leaves 99 uA for Rb2. If now for some reason beta is halved, the base current would become 2 uA. So now 98 uA flows through Rb2. Thatś not much of a difference now is it ?
Compare that to the situation where Ib = 1 uA but Rb1 provides only 2 uA so for Rb2 thereś only 1 uA left. Now if beta halves there would be zero current left for Rb2. That would not actually happen of course, it would settle somewhere in the middle.
But notice how by "wasting" current through Rb1, Rb2 I can basically ignore what happens to the base current and therefore beta as well.
For small signals adding Rb2 also has an advantage as Rb2 with Rb1 forms a voltage divider controlling how much of the output signal is fed-back.
Without Rb2 there will only be the internal small signal input resistance of the NPN, it has value beta/gm. Note how beta is in there again !
By adding Rb2 and making it much lower value than beta/gm Rb2 "takes over" and allows us to have more control and also making the influence of beta smaller.