First, I am not very experienced with hardware, I did a few projects with not more than 30 components, usually less. And in the software field, I'm more at home in the lower-range embedded area (Cortex M & similiar), and Cortex A based singleboard devices.
> The evaluation boards of such high speed ADCs are controlled by FPGAs, that I am trying to avoid them for a beginner like me.
A good idea, I think. Both because of complexity and costs. The "FPGA" is usually a proprietary unit with specialized hardware and a common processor core to control it.
> I will try to explore the DSP way. I read they are fast math operator. ...
Like with general purpose MCUs, there is a broad range of DSP, depending on the target application. I think TI and AD are relatively strong in this field. And they can get quite complex, too.
But there advantage is mostly in architecture, structure, and consequently instruction set. DSPs are more optimised toward high data throughput rates.
Most have two separate RAM blocks at separate internal busses, allowing for simultaneous transfers either by core instructions or DMA. And most of them are VLIW architectures, encoding several instructions in each (long) instruction word. Interrupt latency is another important feature.
But you need to consider the toolchain and debug support as well. Because of lacking experience, I can't give any recommendations here. But you could do a search, skim through a few datasheets, and check a few DSP communities.
But coming back to the LPC-Link and the LPC4370 ...
Are you aware of the Labtool board : https://www.embeddedartists.com/products/labtool/
AFAIK it uses the evaluation-feature of the LPC4370 on the LPC-Link, on a board that implements scope-like features.
And if I'm not mistaken, the LPC4370 MCU on the LPC-Link has an internal JTAG connection, i.e. can be debugged by an external debug pod.