Like the thread above I think wheels are the way to go.
I have a British eagle 501 touring bike (my old student bike actually!) And with a dash of paint and better wheels - I put a pair of shimano r550 wheels on it with 25mm pro4 wheels and it's brilliant fast yet secure and bomb proof!
As for being able to keep up with 'modern' it all depends on you - its not always about the bike
I managed to build this up for well under £500 (prepandemic!). Its built from the top Columbus steel tubeset of the day and runs some good solid Shimano 105/ Ultegra 10speed sourced from a car-boot sale
To throw another possible route - there are numerous examples of 15odd year old bikes out there that do not fall in the 'retro/vintage/modern' category, yet can be a blast to ride. For an example, this is an aluminium/carbon mixed Claud Butler Vicenza, full Campaq Xenon 2x9 groupset, shod on CXP22 rims (the massive branding letters hidden under some 3M decals I had laying around).
Will change to flat bars and add mudguards (can run 32mm on it). It is in excellent condition and pretty comfortable with carbon forks and rear/seat stay combo and 25mm instead of 22mm tyres it had on.
Paid under £100........ just because it is not 'desirable'