Why is it difficult to lower the overdrive voltage (Vgs-Vt)?
You ask about lowering the overdrive voltage but what you actually mean is making the transistor work equally well with a lower overdrive voltage.

The overdrive voltage isn't a property of the transistor itself, it is the voltage that is "left over" when subtracting \$V_t\$ from the supply voltage \$V_{DD}\$:

\$V_{overdrive} = V_{DD} - V_t\$

If we decrease \$V_{DD}\$ and not change \$V_t\$ then \$V_{overdrive}\$ will become smaller.

A smaller \$V_{overdrive}\$ will result in smaller drain currents which cause slower voltage changes (as the input capacitance of the next stage will take longer to charge/discharge) resulting in a slower circuit.

How much current a MOSFET conducts is relative to the parameter \$K\$ which is:

\$K=\mu C_{ox}\frac WL\$

What are the ways that we can increase \$K\$ to get more current to compensate for the lower \$V_{overdrive}\$?

\$\mu\$ is the mobility of Silicon and not something we can easily change (if we'd change it that would need a different, often more expensive, manufacturing technology)

Can we increase \$C_{ox}\$ ? If we use a thinner gate oxide then \$C_{ox}\$ would increase but at the same time the threshold voltage \$V_t\$ would decrease as well. That would result in higher leakage currents.

We could try to compensate for the smaller drain current by making the transistors wider so that we get back to the same drain currents as before. In the equation for \$K\$ that means increasing \$W\$.

However, a wider transistor is also larger and therefore has more input capacitance (mainly gate capacitance) which means the larger drain currents do not help as the capacitance will increase with the same factor as the width of the transistor. In the end there is nothing to be gained this way.

We could make \$L\$ smaller, however, in high speed logic circuits we should already be using the smallest value of \$L\$ that is possible. So we cannot make \$L\$ any smaller.

Conclusion:

We cannot simply make \$V_{overdrive}\$ smaller as it is a value that results from the compromise that has already been made between supply voltage, speed, power consumption, leakage currents etc.