Why it is forbidden. Uvdin dechol? But on weekdays I have no free time for such pleasure activities. We have an eruv in my town.
It is assur for a few reasons
For one as you mentioned because of uvdin dechol, is not subjective. Rather it's defined as something that is primarily a non Shabbos activity. So just because you have no time to do it during the week doesn't remove the technical defintion or classification of the act of biking.
Secondly it's forbidden to ride it in a place where there is no eruv.
Furthermore it is forbidden because it's common to fix the chain or the spikes if it breaks. So it's forbidden for that fear.
See Rav Ovadia in Yalkut Yosef as well:
יש להורות למעשה שלא לרכוב על אופניים [המיועדים לאנשים גדולים] בשבת. ואין לפרוץ גזר בזה. [ילקוט יוסף שבת כרך ה עמוד נו]. ונכון מאד להחמיר שלא לנסוע בשבת על אופניים, אפילו כשרוצה ללכת לדבר מצוה, וכמו שהסכימו כן הרבה אחרונים משום עובדין דחול. ובאיסור שבת החמורה ראוי לחוש לדבריהם להחמיר. [יביע אומר ח''י חאו''ח סימן נד אות יב, ושם בהערות על רב פעלים ח''א בתחלה]. אבל אופניים קטנות תלת אופן [שלש גלגלים] המיועדות למשחק הילדים, מותר להניח לילדים שישחקו בהם בשבת, באופן שאין שם בטריה ולא נדלקת מנורה. [ילקו''י דיני חינוך קטן עמוד רטז]
See here as well with a bit more clarification into the opinion of Rav Ovadia:
Maran Harav Ovadia Yosef Shlit”a (in his Chazon Ovadia-Shabbat Part 4 page 43, as well as in his Responsa Yabia Omer Volume 10 in his comments on the Responsa Rav Pe’alim) agrees that riding a bicycle on Shabbat is prohibited based on what the Gemara in Masechet Shabbat (113a) writes, “’If you shall turn away your foot because of the Shabbat-by not making your ways,’ your mode of walking on Shabbat should not be like your mode of walking during the weekdays.” The Poskim derive from this Gemara that one may not run on Shabbat, as we shall discuss in the following Halacha. > > Thus, since bicycle-riding is meant for traveling a long distance which is not the usual way of walking on Shabbat, it is forbidden to ride bicycles on Shabbat. He proceeds to bring many sources to defend his opinion, one of which is based on the Gemara in Masechet Beitzah (25b) which states that one may not go out in a chair on Shabbat. This refers to the custom of an important figure sitting in a chair and being carried around by people to his destination of choice (as is common in some lands in the Far East even today). Our Sages prohibited this practice on Shabbat, for this is not respectful to the Shabbat as this is considered a weekday mode of travel. Maran Shlit”a brings other reasons to be stringent as well. > > Thus, halachically speaking, one should not be lenient to ride a bicycle on Shabbat; even if one is doing so for the purpose of performing a Mitzvah, one should still act stringently regarding this matter. We shall, G-d-willing, discuss the laws regarding bicycle-riding for children in the Halacha that will be published on Thursday.