Not necessarily. I do struggle when playing anything pre-NES though.
Even then, some games just haven't aged well in comparison to what's currently available. Especially the early polygonal stuff... it just feels bare. I enjoy my highly detailed worlds, my modern UI, and my quality of life improvements. I'm spoiled. Lots of retro titles can't really hold my interest, but I don't think it's because they're "too slow".
Gameplay wise, in general, no. It's a game by game basis and I don't really see age as being a factor.
As stated above a lot of genres, RPGs especially, have had "quality of life" tweaks over the years that can speed up how you progress through a game considerably.
I thought you said Old Console Game Show, and I was remembering all the great ones shown here in BC, Canada back in the 90s and also the ones I found out about recently that were playing in the UK. That's a great topic. I know that on a lot of the Game Boy variants and other original hardware; Chip-tune musicians do what's called bit crushing. At least some of them modify their hardware through controlled overclocking/underclocking - modifying the clock speed of the CPU in their console. This is common enough that you can buy tried and tested clock crystals and potentiometers that would allow push-button double-speed or variable speed while playing a real cartridge on a real Game Boy. It's usually used to allow more octaves in the low dynamic range of the original sound generation hardware through underclocking; The Game Boy is notorious for it's inability to output any bass frequencies despite the audio chip's waveform and PCM capability being ideal for really textural sound generation. Double speed and half speed allow more octaves in the music if used right, as the game boy has a fairly limited dynamic range when composing music. I recall many musical covers on unmodified consoles needed to compose the music an octave higher to achieve all the notes.
Because of this well-known and widely used hack it's easily possible (for someone who can do simple hardware modification) to add in a speed control on a piece of old console hardware. I don't think it's too far a stretch that I could source a double-speed quartz and the other required hardware to make an inconspicuous small tact button trigger as a fast-forward button like the emulators allow. I even know a couple of greatly skilled people who would gladly install it for a reasonable price.
Yes, I share this experience to the point that I cannot enjoy old RPGs without having a FF-speed button mapped to an unused trigger on my wireless controller in an emulator. I didn't like that I needed it to have fun in my old favorites, and treasuring the time I had, I simply stopped playing turn-based or overly grindey RPGs. I simply switched to action RPGs made for the same systems - there's an amazing amount of action-combat and still true RPGs (Most with at least a decent storyline, tho often only in the manual and sparse cutscenes) available for all of the old systems. When fan translations are taken into account the ARPG selection is vast for all of the post-atari-systems and I wish I had more time to enjoy some of the gems available.
If you only want to play Pokemon on original hardware, I am sure you can get a double speed or more button installed for a reasonable price. If you can't find someone willing I'd even do it. I think I have a double-speed quartz in a ziploc somewhere... I may take 6+ moz so I don't recommend my service.
If you want to have that same experience on the cheap, with a better screen (I reject "better" screen when I can play on original or SD CRT) I can verify from experience that you can emulate it all fairly accurately on all of the later handheld systems, and the emulator authors mostly all allow that same ability, usually with a speed selection, able to be mapped to a trigger button. IIRC this is available on GBA, DS, DSI, 3DS, N3DS, PSP, PSP-GO, PS-VITA, all Android devices, most Apple devices etc... Yeah, I really miss when I had the time to really properly enjoy a great 8 or 16-bit JRPG - the grinding was very compelling. The reward was visceral. Taking on the great unbeatable obstacles with far superior earned power was so awesome. I'm probably not going to play through one again until I'm unable to do other stuff.
Some badly programmed older games run slower, but that’s partly because they can’t sustain a good frame rate for faster action. A lot of older arcade games had slow player movement but the enemies would keep speeding up until they were virtually teleporting. And, notably, most GB games are pretty slow, to minimize the impact of ghosting.