Hi Jothar,
Lakewood is born of the yeshiva community and was not a modern community, if you meant MO by that. Even the Old Shul on Monmouth, probably over 100 years old now, was totally yeshivish in origin and orientation, even though Jews of all stripes could always be found in Lakewood.
At the same time, however, Lakewood always included people on all levels who get/got along well.
With regards to the possible implications of your statement that Rav Kotler ZT”L, went to a “day school” as a choice for a modern day school as one may find today, the legendary Bezalel Hebrew Day School, was practically the only game in town. Those who went there did not go there “for a modern experience versus a more right/black hat experience”. That notion would be incorrect. The school had as deep a limud kodesh as you would have found anywhere in the world.
Today, Lakewood is a fantastic mix of all kinds of yidden. But the implication of it being “less Jewish” could not be further from the truth. The standards put forth there could not be higher. Thank heaven for Lakewood. There is no place like it, especially on Shabbos.