Lets *NOT* consider original (and long obsolete) 7400 series TTL logic, as the output isn't rail to rail and the input characteristic is asymmetric, which makes the though experiment harder. Instead, lets consider 74HC00 series logic - CMOS logic with the same pinout as 7400 TTL, and negligible input current requirements so at low frequencies fanout is only limited by the total capacitance loading the signal, and one's max. risetime requirement.
Consider your diode OR gate, driven by a 5V 74HC CMOS output. Take the diodes' Vf drop as 0.7V, and you'll see that after one stage of diode OR gates you'll have a logic '1' level of 4.3V. That's still OK for driving 74HC logic as its thresholds for logic '1' and '0' are 70% and 30% of Vcc respectively to guarantee the input is recognised as the correct level, so the input needs to be above 3.5V. Add another stage of diode OR gates and the voltage drops to 3.6V - that's still just OK to drive 74HC logic, but there's not much noise margin - it would only take a small disturbance for the signal to enter the indeterminate region. Its therefore obvious that if you are using diode logic, the signal needs to be amplified every stage or two to regenerate the required rail-to-rail output levels.
The situation is worse if you mix diode AND and OR gates. Consider a diode OR driving a diode AND. For a logic '0', the first gate only drives the second passively via its pulldown, which forms a potential divider with the second gate's pullup, so if they are equal, you'll get a output voltage of approx 2.8V for logic '0' - clearly invalid and unusable.
The situation is worse for 7400 series and other BJT 74xx00 'TTL' logic series as the output swing (as a proportion of Vcc) is much less than 74HC and other CMOS logic families, and a significant current is required to pull an input down to a valid logic '0' level.
It should be noted that the input stages of 74LS and some other second/third generation BJT Schottky 'TTL'* logic families typically use diode logic internally if an AND/NAND function is required at the input.
* actually not TTL as they don't use multiple emitter input transistors, but TTL level input and output compatible.