In some cases you can do that, but often better technology just leads to weaker signals. For example, the microphone capsule that is in a 1960s "black" phone is sensitive and requires no amplifier, but is large and expensive to manufacture. An electret microphone has a signal that is so weak it needs the amplifier built into the capsule but it's cheap, very small and quite good. Ribbon microphones are very, very good but have extremely low output voltage (fortunately also with a low source impedance).
If you want to get a temperature signal that is mV/degree rather than uV/degree you could build a thermopile with 1,000 junctions but it would suck 1000x the heat away from whatever it is that you are trying to measure (all other things being equal), so it would be a terrible thermal sensor. Even before amplifiers were practical, the output of a single junction was typically used for measurement and control purposes (using finely balanced galvanometers).
A photoconductive cell has large output but it contains cadmium and is undesirable for that reason (and others). A photodiode has a weak output, but is cheap to manufacture, small, easily amplified and is faster and far more stable.
At some point you may run into the limits of physics such as Johnson-Nyquist noise (more generally the fundamental fluctuation-dissipation theorem) - the signal is so weak that you cannot achieve the required signal to noise ratio even with the finest amplifier practical or even possible, and in that case you may have to make the sensor larger and heavier to achieve the required performance.. for example with accelerometers and gyros. The tiny ones tend to be rather noisy and drifty.
So sometimes you have to make the transducer big and heavy and expensive (see LIGO), for example, but if you can, economics tends to lead to making the transducer small and cheap and less sensitive and adding an amplifier. More so as amplifiers tend to get better and cheaper over time and material costs tend to rise over time. An engineer that uses a $100 more expensive transducer to avoid a 5 cent amplifier for no good reason won't likely be long employed. As one data point, a low-drift DC amplifier from 1972 can be outperformed by couple orders of magnitude in a device that today costs a couple orders of magnitude less in real terms (~10,000:1 improvement in price/performance).